Arithmetic Sequences and Arithmetic Series - Basic Introduction

Summary

This educational video script explores arithmetic and geometric sequences, distinguishing between the two by their patterns of difference and ratio. It explains how to calculate means, both arithmetic and geometric, and introduces formulas for finding the nth term and the sum of sequences and series. The script also provides examples of identifying sequences and series, whether finite or infinite, and demonstrates how to apply formulas to find specific terms and sums in arithmetic sequences. It further illustrates the process with practice problems, reinforcing the concepts taught.

Takeaways

• 🔢 Arithmetic sequences have a common difference; for example, 3, 7, 11, 15.
• 📈 Geometric sequences have a common ratio; for example, 3, 6, 12, 24.
• ➕ In arithmetic sequences, you add a constant to get the next term.
• ✖️ In geometric sequences, you multiply by a constant to get the next term.
• 📊 The arithmetic mean of two numbers in a sequence is their average, like (3+11)/2=7.
• 🔍 The geometric mean of two numbers in a sequence is the square root of their product, like sqrt(3*12)=6.
• 🔢 The formula for the nth term of an arithmetic sequence is a_n = a_1 + (n-1)*d.
• 📈 The formula for the nth term of a geometric sequence is a_n = a_1 * r^(n-1).
• ➗ Partial sums of arithmetic and geometric sequences have specific formulas.
• 🔄 Sequences are lists of numbers; series are sums of sequences.

Q & A

What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers with a common difference between consecutive terms. For example, the sequence 3, 7, 11, 15, 19, 23, and 27 has a common difference of 4.

What is the difference between an arithmetic sequence and a geometric sequence?

An arithmetic sequence has a common difference between consecutive terms, while a geometric sequence has a common ratio by which each term is multiplied by to get the next term. For example, the sequence 3, 6, 12, 24, 48, 96 is geometric with a common ratio of 2.

How do you calculate the arithmetic mean of two numbers within an arithmetic sequence?

The arithmetic mean of two numbers within an arithmetic sequence is found by adding the two numbers and dividing by two. For instance, the mean of 3 and 11 is (3 + 11) / 2 = 7.

What is the formula to find the nth term of an arithmetic sequence?

The formula to find the nth term (a_sub_n) of an arithmetic sequence is a_sub_n = a_sub_1 + (n - 1) * d, where a_sub_1 is the first term and d is the common difference.

How do you find the sum of the first n terms of an arithmetic sequence?

The sum of the first n terms (S_sub_n) of an arithmetic sequence is found using the formula S_sub_n = (a_sub_1 + a_sub_n) / 2 * n, where a_sub_1 is the first term and a_sub_n is the nth term.

What is the difference between a sequence and a series?

A sequence is a list of numbers in a specific order, while a series is the sum of the numbers in a sequence. For example, the list 3, 7, 11, 15 is a sequence, and 3 + 7 + 11 + 15 is a series.

How do you determine if a sequence is finite or infinite?

A sequence is finite if it has a definite end, and infinite if it continues indefinitely, often indicated by ellipsis (...) at the end.

What is the formula to calculate the nth term of a geometric sequence?

The formula to find the nth term (a_sub_n) of a geometric sequence is a_sub_n = a_sub_1 * r^(n-1), where a_sub_1 is the first term and r is the common ratio.

How do you calculate the sum of the first n terms of a geometric sequence?

The sum of the first n terms (S_sub_n) of a geometric sequence is calculated using the formula S_sub_n = a_sub_1 * (1 - r^n) / (1 - r), where a_sub_1 is the first term and r is the common ratio, provided r ≠ 1.

What is the difference between the arithmetic mean and the geometric mean of two numbers?

The arithmetic mean of two numbers is the average found by adding the numbers and dividing by two. The geometric mean is the square root of the product of the two numbers. For example, the arithmetic mean of 3 and 12 is (3 + 12) / 2 = 7.5, while the geometric mean is √(3 * 12) = √36 = 6.

How can you identify whether a sequence is arithmetic or geometric by looking at its terms?

A sequence is arithmetic if there is a common difference between consecutive terms. It is geometric if there is a common ratio by which consecutive terms are multiplied to get the next term.

What is the sum of the first 300 natural numbers?

The sum of the first 300 natural numbers can be calculated using the formula for the sum of an arithmetic series: S_sub_n = n * (a_sub_1 + a_sub_n) / 2, where n = 300, a_sub_1 = 1, and a_sub_n = 300. The sum is 300 * (1 + 300) / 2 = 45150.

Outlines

00:00

🔢 Arithmetic vs. Geometric Sequences

This paragraph introduces the concepts of arithmetic and geometric sequences, providing examples for each. It explains the difference between a common difference in arithmetic sequences and a common ratio in geometric sequences. The paragraph also discusses the calculation of arithmetic and geometric means, illustrating how to find the average of two numbers in an arithmetic sequence and the middle term in a geometric sequence.

05:01

📚 Formulas for Sequences and Series

The second paragraph delves into the formulas for finding the nth term and the partial sum of both arithmetic and geometric sequences. It demonstrates how to calculate any term in an arithmetic sequence using the first term and the common difference, and similarly for a geometric sequence using the first term and the common ratio. The paragraph also explains how to find the sum of the first n terms of each sequence type, providing formulas and examples.

10:03

📉 Summation of Sequences and Series

This paragraph continues the discussion on sequences and series, focusing on the process of summing the terms of a sequence to form a series. It differentiates between finite and infinite sequences and series, providing examples for clarity. The paragraph also emphasizes the importance of understanding the difference between a sequence (a list of numbers) and a series (the sum of those numbers in a sequence).

15:04

🔍 Identifying Sequences and Series

The fourth paragraph presents practice problems to identify whether a given list of numbers is a finite or infinite sequence or series and whether it is arithmetic, geometric, or neither. It provides a method to determine the common difference or ratio and to classify the sequence accordingly. The paragraph also summarizes the answers to the practice problems, categorizing each into the correct type of sequence or series.

20:04

✍️ Writing Terms of Defined Sequences

This paragraph involves writing the first few terms of sequences defined by specific formulas. It explains how to find the initial terms of a sequence by substituting values into the given formula and how to identify the pattern to continue the sequence. The paragraph also covers recursive formulas, showing how to derive subsequent terms based on previous ones.

25:04

📘 Explicit Formulas for Sequences

The sixth paragraph discusses how to write explicit formulas for arithmetic sequences, given the first term and the common difference. It provides a step-by-step process for creating the general formula and applies this to sequences involving fractions by separating the numerator and denominator into distinct sequences. The paragraph illustrates this with examples, ensuring the formulas accurately represent the given sequences.

30:05

📊 nth Term Formulas and Summation

This paragraph focuses on writing nth term formulas for arithmetic sequences and calculating the value of specific terms and the sum of the first n terms. It provides the general formula for an arithmetic sequence and demonstrates its application to find the tenth term and the sum of the first ten terms. The paragraph also includes examples with different sequences to illustrate the process.

35:06

🧩 Sum of Natural Numbers and Even Numbers

The eighth paragraph addresses the calculation of the sum of the first 300 natural numbers and the sum of all even numbers from 2 to 100 inclusive. It uses the formula for the sum of an arithmetic series and provides a step-by-step solution for each case. The paragraph also explains how to determine the number of terms in a sequence when the last term is known.

40:06

🎯 Summation of Odd Integers and Sequence Calculation

The ninth and final paragraph discusses the sum of odd integers from 21 to 75. It explains how to determine the number of terms in the sequence and how to calculate the sum using the arithmetic series formula. The paragraph provides a clear method for finding the sum of a specific range of numbers in a sequence, emphasizing the importance of identifying the correct number of terms.

Keywords

💡Arithmetic Sequence

An arithmetic sequence is a series of numbers in which each term after the first is obtained by adding a constant difference to the previous term. It is central to the video's theme as it demonstrates the fundamental concept of arithmetic progressions. For example, the sequence 3, 7, 11, 15, 19, 23, and 27 is used in the script to illustrate this concept, where the common difference is 4.

💡Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This concept is juxtaposed with arithmetic sequences in the video to highlight the difference in patterns. The script provides the example 3, 6, 12, 24, 48, 96, 192 to demonstrate a geometric sequence with a common ratio of 2.

💡Common Difference

The common difference in an arithmetic sequence is the constant amount added to each term to get the next term. It is a key element in the formation of arithmetic sequences and is used in the script to explain the pattern within the sequence 3, 7, 11, and so on, where the common difference is 4.

💡Common Ratio

The common ratio in a geometric sequence is the constant factor by which each term is multiplied to obtain the next term. The video script uses the common ratio to explain the pattern in the sequence 3, 6, 12, etc., where the common ratio is 2, indicating that each term is twice the previous one.

💡Arithmetic Mean

The arithmetic mean, often referred to as the average, is calculated by adding a set of numbers and then dividing by the count of numbers. In the context of the video, the arithmetic mean is used to find the middle number in an arithmetic sequence, such as the mean of 3 and 11, which is 7.

💡Geometric Mean

The geometric mean of two numbers is found by multiplying the numbers together and then taking the square root of the product. The script illustrates this by showing that the geometric mean between 3 and 12 is 6, as the square root of 36 (3 * 12) is 6.

💡Nth Term

The nth term of a sequence refers to the specific term in a sequence that is the nth position. The video explains how to find any term in an arithmetic or geometric sequence using formulas. For instance, the formula for the nth term of an arithmetic sequence is a_n = a_1 + (n - 1) * d, where 'd' is the common difference.

💡Partial Sum

The partial sum, denoted as S_n, is the sum of the first n terms of a sequence. The video script discusses how to calculate the partial sum of an arithmetic sequence using the formula S_n = (a_1 + a_n) / 2 * n, and for a geometric sequence, it's S_n = a_1 * (1 - r^n) / (1 - r), where 'r' is the common ratio.

💡Sequence

A sequence is an ordered list of numbers or objects. The video script distinguishes between different types of sequences, such as arithmetic and geometric, and explains the properties that define them, like common differences or ratios.

💡Series

A series is the sum of the terms in a sequence. The script clarifies the difference between a sequence and a series, with the latter being a cumulative total of the numbers in the former. For example, the sum of the arithmetic sequence 3, 7, 11, ..., up to the nth term, is referred to as an arithmetic series.

💡Recursive Formula

A recursive formula is a definition of a sequence where each term is defined in terms of the previous terms. The video script provides examples of recursive formulas and demonstrates how to use them to find subsequent terms in a sequence, such as a_(n) = a_(n-1) + 4.

Highlights

Introduction to the concept of arithmetic sequences and their distinction from geometric sequences.

Explanation of the common difference in arithmetic sequences and common ratio in geometric sequences.

Demonstration of how to calculate the arithmetic mean and its significance in arithmetic sequences.

Illustration of calculating the geometric mean and its relevance to geometric sequences.

Introduction to the formulas for finding the nth term in both arithmetic and geometric sequences.

Practical application of the formula to find the fifth term in an arithmetic sequence example.

Explanation of how to calculate the sixth term in a geometric sequence using the formula.

Clarification on the difference between sequences and series, and their finite and infinite forms.

Tutorial on identifying the type of sequence or series and whether it's finite or infinite.

Method to determine if a sequence is arithmetic, geometric, or neither through pattern recognition.

Practice problems to apply the understanding of sequences and series.

How to write the first four terms of a sequence using a given formula.

Finding the next three terms of an arithmetic sequence given its first few terms.

Writing the first five terms of an arithmetic sequence given the first term and common difference.

Solving recursive formulas to determine terms in a sequence.

Developing a general formula for sequences shown, including handling sequences with fractions.

Writing a formula for the nth term of arithmetic sequences and calculating specific terms.

Calculating the sum of the first 10 terms of an arithmetic sequence using the sum formula.

Finding the sum of the first 300 natural numbers using the arithmetic series sum formula.

Summing all even numbers from 2 to 100 inclusive by identifying the sequence and using the sum formula.

Determining the sum of odd integers from 20 to 76 by calculating the value of n and applying the sum formula.

Transcripts

in this video we're going to focus

mostly on arithmetic sequences

now to understand what an arithmetic

sequence is it's helpful to distinguish

it from a geometric sequence

so here's an example of an arithmetic

sequence

the numbers 3 7 11

15

19

23 and 27 represents an arithmetic

sequence

this would be a geometric sequence 3

6

12

24

48

96 192.

do you see the difference between these

two sequences and do you see any

patterns within them

in the arithmetic sequence on the left

notice that we have a common difference

this is the first term this is the

second term this is the third fourth and

fifth term

to go from the first term to the second

term we need to add four

to go from the second to the third term

we need to add four

and that is known as the common

difference

in a geometric sequence you don't have a

common difference rather you have

something that is called the common

ratio to go from the first term to the

second term

you need to multiply by two

to go from the second term to the third

term you need to multiply by two again

so that is the r value that is the

common ratio

so in an arithmetic sequence the pattern

is based on addition and subtraction

in a geometric sequence the pattern is

based on multiplication and

division now the next thing that we need

to talk about

is

the mean

how to calculate the arithmetic mean and

the geometric mean

the arithmetic mean is basically the

average of two numbers

it's a plus b divided by two

so when taking

an arithmetic mean of two numbers within

an arithmetic sequence let's say if we

were to take

the mean of three and eleven

we would get the middle number in that

sequence in this case we would get 7.

so if you were to add 3 plus 11

and divide by 2 3 plus 11 is 14 14

divided by 2 gives you 7.

now let's say if we wanted to find the

arithmetic mean between 7

and 23 it's going to give us the middle

number

of that sequence which is 15.

so if you would add up 7

plus 23 divided by 2

7 plus 23 is 30 30 divided by 2 is 15.

so that's how you can calculate the

arithmetic mean and that's how you can

identify it within

an arithmetic sequence

the geometric mean

is the square root

of a times b

so let's say if we want to find the

geometric mean between three and six

it's going to give us the middle number

of the sequence which is

i mean if we were to find the geometric

mean between 3 and 12

we will get the middle number of that

sequence which is 6.

so in this case a is 3 b is 12. 3 times

12 is 36

the square root of 36 is 6.

now let's try another example

let's find the geometric mean

between 6 and 96

this should give us the middle number

24.

now we need to simplify this radical

96 is six times sixteen

six times six is thirty six

the square root of thirty six is six the

square root of sixteen is four

so we have six times four

which is twenty 24.

so as you can see

the geometric mean of two numbers within

the geometric sequence will give us the

middle number in between those two

numbers in that sequence

now let's clear away a few things

the formula that we need to find the nth

term

of an arithmetic sequence is a sub n

is equal to a sub 1

plus n minus 1 times the common

difference d

in a geometric sequence

its a sub n

is equal to a 1

times r

raised to the n minus 1.

now

let's use that equation to get the fifth

term in the arithmetic sequence

so that's going to be a sub 5

a sub 1 is the first term which is three

n is five since we're looking for the

fifth term the common difference

is four in this problem

five minus one is four

4 times 4 is 16 3 plus 16 is 19.

so this formula

gives you any term in the sequence you

could find the fifth term the seventh

term the 100th term and so forth

now in a geometric sequence

we could use this formula

so let's calculate the six term

of the geometric sequence it's going to

be a sub six

which equals a sub one the first term is

three the common ratio is two

and this is going to be raised to the

six minus one

six minus one is five

and then two to the fifth power if you

multiply two five times two times two

times two times two times two

so we can write it out

so this here that's four

three twos make eight four times eight

is thirty two

so this is three

times thirty two

three times thirty is ninety three times

two is six

so this will give you

ninety six

so that's how you could find the f term

in a geometric sequence

by the way

make sure you have a sheet of paper to

write down these formulas

so that when we work on some practice

problems

you know what to do

now the next thing we need to do is

be able to calculate the partial sum of

a sequence

s sub n is the partial sum of

a series of a few terms

and it's equal to the first term plus

the last term

divided by two times n

for geometric sequence the partial sum s

of n is going to be a sub 1

times 1 minus r raised to the n

over

1 minus r

so let's find the sum

of the first seven terms in this

sequence

so that's going to be s sub 7

that's going to equal the first term

plus the seventh term

divided by 2

times n

where n is the number of terms which is

7.

now think about what this means

so basically to find the sum of an

arithmetic sequence

you're basically taking the average

of the first and the last term in that

sequence and then multiplying it by the

number of terms

in that sequence

because this is basically the average of

3

and 27.

and we know the average or the

arithmetic mean of 3 and 27 that's going

to be the middle number 15.

so let's go ahead and plug this in

so this is 3 plus 27

over 2

times 7.

3 plus 27 is 30 plus 2 i mean well 30

divided by 2 that's 15.

so the average

of the first and last term is 15 times 7

10 times 7 is 70. 5 times 7 is 35

so this is going to be 105.

that's the sum of the first seven terms

and you can confirm this with your

calculator if you add up three plus

seven

plus eleven

plus fifteen

plus 19 plus 23

and then plus 27

and that will give you s of 7 the sum of

the first seven terms

go ahead and add up those numbers

if you do you'll get 105.

so that's how you can confirm your

answer

now let's do the same thing with a

geometric sequence

so let's get the sum

of the first six terms

s sub six

so this is going to be three

plus six

plus twelve

plus twenty four

plus 48

plus 96

so we're adding the first six terms

now because it's not many terms we're

adding we can just simply plug this into

our calculator

and we'll get 189

but now let's confirm this answer using

the formula

so s sub 6 the sum of the first six

terms

is equal to the first term a sub one

which is three

times one minus r r is the common ratio

which is two

raised to the n

n is six

over one minus r or 1 minus 2.

so i'm going to work over here since

there's more space

now 2 to the 6

that's going to be 64. if you recall 2

to the fifth power was 32 if you

multiply 32 by 2 you get 64.

so this is going to be 1 minus 64.

and 1 minus 2 is negative 1.

so this is 3 times

1 minus 64.

is negative 63.

so we could cancel the two negative

signs a negative divided by a negative

will be a positive

so this is just 3 times 63.

3 times 6 is 18. so 3 times 60 has to be

180

and then 3 times 3 is 9 180 plus 9

adds up to 189

so we get the same answer

now what is the difference between

a sequence and a series

i'm sure you heard of these two terms

before but what is the difference

between them

now we've already considered what

an arithmetic sequence is

a sequence is basically a list of

numbers

so that's a sequence

a series is the sum

of the numbers in a sequence

so this here is

an arithmetic sequence

this

is an arithmetic series because it's the

sum of an arithmetic sequence

now what we have here is a sequence but

it's a geometric sequence as we've

considered earlier

this

is a geometric series it's the sum of a

geometric sequence

now there are two types of sequences and

two types of series

you have a finite sequence

and an infinite sequence and it's also a

finite series and an infinite series

this sequence is finite it has a

beginning and it has an n

this series is also finite it has a

beginning and it has an end

in contrast if i were to write 3 7

11

15 19 and then dot dot dot

this would be

an infinite sequence

the presence of these dots tells us that

the numbers keep on going to infinity

now the same is true for a series

let's say if i had

three plus seven

plus eleven plus fifteen plus nineteen

and then plus dot dot dot dot

that would also be

an infinite series

so now you know the difference between a

finite series and an infinite series

now let's work on some practice problems

describe the pattern of numbers shown

below is it a sequence or series

is it finite or infinite

is it

arithmetic geometric or neither

so let's focus on if it's a sequence or

series first

part a

so we got the numbers 4 7 10 13 16 19.

we're not adding the numbers we're

simply making a list of it so this is

a

sequence the same is true for part b

we're simply listing the numbers so

that's a sequence

in part c we're adding a list of numbers

so since we have a sum this is going to

be a series

d is also a series

e

that's a sequence

for f we're adding numbers so that's a

series and the same is true

for g

so hopefully this example helps you to

see the difference between a sequence

and a series

now let's move on to the next topic is

it finite or is it infinite

to answer that all we need to

do is identify if we have a list of dots

at the end or not

here this ends at 19. so that's a finite

sequence

the dots here tells us it's going to go

forever so this is an infinite

sequence

this one we have the dots so this is

going to be an infinite series

this ends at 162 so it's finite so we

have a finite series

this is going to be an infinite sequence

next we have an infinite series

and the last one is a finite series

now let's determine if we're dealing

with

an arithmetic geometric

or neither sequence or series

so we're looking for a common difference

or a common ratio

so for a

notice that we have a common difference

of three four plus three is seven

seven plus three is ten

so because we have a common difference

this is going to be

an arithmetic sequence

for b going from the first number to the

second number we need to multiply by two

four times two is eight eight times two

is sixteen

so we have a common ratio

which makes this sequence geometric

for answer choice c

going from five to nine that's plus four

and from nine to thirteen that's plus

four so we have a common difference

so this is going to be not an arithmetic

sequence

but an arithmetic series

for answer choice d going from two to

six we're multiplying by three and then

six times three is eighteen

so that's a geometric

a geometric series

now for e

going from 50 to 46 that's a difference

of negative 4 and 46 to 42 that's the

difference of negative four so this

is arithmetic

for f

we have a common ratio of four three

times four is twelve

twelve times 4 is 48

and if you're wondering how to calculate

d and r

to calculate d take the second term

subtracted by the first term

7 minus 4 extreme or you could take the

third term subtracted by the second 10

minus 7 is 4.

in the case of f if you take 12 divided

by 3 you get 4.

48 divided by 12 you get 4. so that's

how you can calculate the common

difference or the common ratio

is by analyzing the second term with

respect to the first one

so since we have a common ratio

this is going to be geometric

for g if we subtract 18 by 12 we get a

common difference of positive six

24 minus 18

gives us the same common difference of

six

so this is going to be

arithmetic

so now let's put it all together let's

summarize the answers

so for part a what we have is a finite

arithmetic sequence

part b

this is an infinite geometric sequence

c

we have an infinite arithmetic series

d

is a finite geometric series

e is an infinite arithmetic sequence

f

is an infinite geometric series

g is a finite arithmetic series

so we have three columns of information

with two different possible choices

thus two to the third is eight which

means that we have eight different

possible combinations

right now i have seven out of the eight

different combinations the last one is a

finite

geometric sequence which i don't have

listed here

so now you know how to

identify whether you have a sequence or

series if it's a rhythmical geometric

and if it's finite or infinite

number two

write the first four terms of the

sequence defined by the formula a sub n

is equal to three n minus seven

so the first thing we're going to do is

find the first term

so we're going to replace n with one

so it's going to be three minus seven

which is negative 4.

and then we're going to repeat the

process we're going to find the second

term a sub 2.

so it's 3 times 2 minus 7

which is negative 1.

next we'll find a sub 3.

three times three is nine minus seven

that's two

and then the fourth term a sub four

that's going to be twelve minus seven

which is five

so we have a first term of negative four

then it's negative one

two

five

and then the sequence can continue

so the common difference in this problem

is positive three

going from negative one to two if you

add three you'll get two

and then two plus three is five

but this is the answer for the problem

so

this is

those are the first four terms of the

sequence

number three

write the next three terms of the

following arithmetic sequence

in order to find the next three terms we

need to determine the common difference

a simple way to find the common

difference is to subtract the second

term by the first term

22 minus 15 is 7.

now just to confirm we need to make sure

that

the difference between the third and the

second term is the same

29 minus 22 is also

seven

so we have a common difference of seven

so we could use that to find in the next

three terms

so 36 plus 7 is 43

43 plus 7 is 50

50 plus 7 is 57

so these are the next three terms of the

arithmetic sequence

here's a similar problem but presented

differently

write the first five terms of an

arithmetic sequence

given a one and d

so we know the first term is 29 and the

common difference is negative four

so this is all we need to write the

first five terms if the common

difference is negative four

then the next term is going to be 29

plus negative four which is 25

25 plus negative four or 25 minus 4 is

21

21 minus 4 is 17

17 minus 4 is 13.

so that's all we need to do in order to

write the first five terms

of the arithmetic sequence given this

information

number five

write the first five terms of the

sequence

defined by the following recursive

formulas

so let's start with the first one part a

so we're given the first term what are

the other terms

when dealing with recursive formulas

we need to realize is that you get the

next term by plugging in the previous

term

so let's say n is 2.

when n is two this is a sub two

and that's going to equal a sub n minus

one two minus one is one so this becomes

a sub one plus four

so the second term is going to be the

first term 3

plus 4 which is 7.

so we have 3 as the first term 7 as a

second term so now let's find the next

one

so let's plug in 3 for n so this becomes

a sub 3

the next one this becomes a sub 3 minus

1 or a sub 2

plus 4.

so this is seven

plus four which is eleven

at this point we can see that we have an

arithmetic sequence with a common

difference of four so to get the next

two terms we could just add four it's

going to be 15 and 19.

so that's it for part a

so when dealing with recursive formulas

just remember you get your next term by

using the previous term

now for part b it there's going to be a

little bit more work

so plugging in n equals 2

we have the second term

it's going to be 3 times the first term

plus 2.

the first term is two

so three times two is six plus two that

gives us eight

so now let's plug in n equals three

when n is 3 we have this equation a sub

3 is equal to 3 times a sub 2 plus 2.

so we're going to take 8 and plug it in

here to get the third term

so it's 3 times 8 plus 2

3 times 8 is 24 plus 2 that's

26.

now let's focus on the fourth term when

n is 4. so this is going to be a sub 4

is equal to 3 times

a sub 3 plus 2.

so now we're going to plug in 26 for a

sub 3.

so it's 3 times 26 plus 2.

3 times 26 is 78 plus 2 that's going to

be 80.

now let's focus on the fifth term

so a sub 5 is going to be 3 times a sub

4 plus 2

so that's 3 times 80 plus 2

3 times 8 is 24 so 3 times 80 is 240

plus 2 that's going to be 200

so the first five terms are 2

8

26

80

and

242 so this is neither

an arithmetic sequence nor is it a

geometric sequence

number six

write a general formula

or explicit formula which is the same

for the sequences shown below

in order to write a general formula or

an explicit formula

all we need is the first term and the

common difference

if it's an arithmetic sequence which for

part a

it definitely is

so if we subtract 14 by 8 we get 6 and

if we subtract 20 by 14 we get 6.

so we can see that the common difference

is positive 6

and the first term

is 8.

so the general formula is a sub n is

equal to a sub 1

plus n minus 1 times d

so all we need is the first term and the

common difference

and we can write a general formula or an

explicit formula

the first term is eight

d is six

now what we're going to do is we're

going to distribute six to n minus one

so we have six times n which is six n

and then this will be negative six

next we need to combine like terms

so eight plus negative six or eight

minus six that's going to be positive

two

so the general formula is six n

plus two

so if we were to plug in one

this will give us the first term eight

six times one plus two is eight

if we were to plug in four

it should give us the fourth term twenty

six

six times four is twenty four

plus two that's twenty six

so now that we have the explicit formula

for part a what about the sequence in

part b

what should we do if we have fractions

if you have a fraction like this or a

sequence of fractions and you need to

write an explicit formula

try to separate it into two different

sequences

notice that we have an arithmetic

sequence if we focus in the numerator

that sequence is

two

three

four

five and six

for the denominator we have the sequence

three five seven

nine eleven

so for the sequence on top the first

term is two and we can see that the

common difference is one

the numbers are increasing by one

so using the formula a sub n is equal to

a sub one plus n minus one times d

we have that a sub one is 2

and d is 1.

if you distribute 1 to n minus 1 you're

just going to get n minus 1.

so we can combine 2 and negative 1

which is positive 1.

so we get the formula n plus one

and you could check it when you plug in

one one plus one is two so the first

term is two

if you were to plug in

five

five plus one is six that will give you

the fifth term which is six

now let's focus on the sequence of the

denominators

the first term is three the common

difference we could see is two

five minus three is two

seven minus five is two

so using this formula again

we have a sub n is equal to a sub one a

sub one is three plus

n minus one times d d is two

so let's distribute two to n minus one

so that's gonna be two n minus two

and then let's combine like terms three

minus two is positive one

so a sub n is going to be two n

plus one

so if we want to calculate the first

term

we plug in one for n two times one is

two plus one it gives us

three

if we wanna calculate the fourth term

and it's four

two times four is eight plus one it

gives us nine

so you always want to double check your

work to make sure that you have the

right formula

so now let's put it all together

so we're going to write a sub n

and we're going to write it as a

fraction the sequence for the numerator

is n plus one

the sequence for the denominator is two

n plus one

so this right here

represents

the sequence

that corresponds to what we see in part

b

and we can test it out let's calculate

the value of the third term

so let's replace n with three it's going

to be three plus one

over two

times three plus one

three plus one is four

two times three is six plus one that's

seven so we get four over seven

if we wish to calculate the fifth term

it's going to be five plus one

over two

times five plus one

five plus one is six

two times five is ten plus one

that's 11.

and so anytime you have to write an

explicit formula given a sequence of

fractions

separate the numerator and the

denominator into two different sequences

hopefully they're both arithmetic

if it's geometric you may have to look

at another video that i'm going to make

soon on geometric sequences

but

break it up into two separate sequences

and then write the formulas that way and

then put the two formulas in a fraction

and that's how you can get the answer

number seven

write a formula for the nth term of the

arithmetic sequences shown below

so writing a formula for the f term is

basically the same as writing a general

formula for the sequence or an explicit

formula

so we need to identify the first term

which we could see as 5

and the common difference

14 minus 5 is 9

23 minus 14 is 9 as well

so once we have these two we can write

the general formula

so let's replace the first term a sub 1

with 5.

and let's replace d with nine

now let's distribute nine to n minus one

so we're gonna have nine n minus nine

next let's combine like terms

so it's going to be 9n and then 5 minus

9 is negative 4.

so this

is the formula for the nth term of the

sequence

now let's do the same for part b

so the first term is 150

the common difference is going to be 143

minus 150

which is negative seven to confirm that

if you subtract 136 by 143 you also get

negative seven

now let's plug it into this formula to

write the general equation

so a sub n is going to be 150

plus

n minus 1 times d which is negative

seven

so let's distribute negative seven to n

minus one

so it's going to be 150 minus seven n

and then negative seven times negative

one that's going to be positive 7.

so a sub n is going to be negative 7n

plus 157

or you could just write it as

157

minus 7n

so that is the formula for the nth term

of the arithmetic sequence

now let's move on to part b

calculate the value of the tenth term

of the sequence

so we're looking for a sub 10. so let's

plug in 10 into this equation

so it's gonna be nine

times ten

minus four

nine times ten is ninety

ninety minus four is eighty-six

so that is the tenth term

of the sequence in part a

for part b

the tenth term is going to be 157

minus seven times ten

seven times ten is seventy one fifty

seven

minus seventy

is going to be eighty seven

now let's move on to part c

find the sum of the first 10 terms

so in order to find the sum we need to

use this formula

s sub n is equal to the first term plus

the last term divided by 2

times the number of terms

so if we want to find the sum of the

first 10 terms we need a sub 1 which we

know it's 5.

a sub n n is 10 so that's a sub 10

the tenth term is 86

divided by 2

times the number of terms which is 10.

5 plus 86 is 91. 91 divided by 2

gives us an average of 45.5 of the first

and last number

and then times 10

we get a total sum of 455.

so that is the sum of the first 10 terms

of this sequence

now for part b

we're going to do the same thing

calculate s sub 10

the first term a sub 1 is 150

the tenth term is eighty seven

divided by two times the number of terms

which is ten

one fifty plus eighty seven that's two

thirty seven

divided by 2 that's

118.5

times 10

we get a sum of 11.85

so now you know how to calculate the

value of the m term and you also know

how to find the sum

of

a series

number eight

find the sum of the first 300 natural

numbers

so how can we do this

the best thing we can do right now is

write a series

zero is not a natural number but one is

so if we write a list one plus two plus

three

and this is going to keep on going

all the way to 300

so to find the sum of a partial series

we need to use this equation s sub n is

equal to a sub 1

plus a sub n over 2 times n

now let's write down what we know

we know that a sub 1 the first term is

one

we know n

is 300

if this is the first term this is the

second term this is the third term this

must be the 300th term

so we know n is 300 and

a sub n or a sub 300 is 300

so we have everything that we need to

calculate the sum of the first 300 terms

so it's a sub 1 which is 1 plus a sub n

which is 300

over 2

times the number of terms which is 300

so it's going to be 301 divided by 2

times 300

and that's

45

150.

so that's how we can calculate the sum

of the first 300

natural numbers

in this series

number nine

calculate the sum of all even numbers

from two to one hundred inclusive

so let's write a series

two is even three is odd so the next

even number is four

and then six and then eight

all the way to one hundred

so we have the first term

the second term is four

the third term is sixty

one hundred is likely to be the 50th

term but let's confirm it

so what we need to do is calculate n and

make sure it's 50 and not 49 or 51.

so we're going to use this equation to

calculate the value of n

so a sub n is a hundred

let's replace that with a hundred

a sub one

is two

the common difference

we see four minus two is two six minus

four is two so the common difference

is two in this example

and our goal is to solve for n

so let's begin by subtracting both sides

by two

a hundred minus two is ninety-eight

and this is going to equal two times n

minus one

next we're going to divide both sides by

two

98 divided by two is 49

so we have 49 is equal to n minus one

and then we're going to add one to both

sides so n is 49 plus 1 which is 50.

so that means that

100 is indeed the 50th term so we know

that n

is 50.

so now we have everything that we need

in order to calculate the sum of the

first 50 terms

so let's begin by writing out the

formula first

so the sum of the first 50 of terms is

going to be the first term which is 2

plus a sub 50 the last term which is 100

divided by 2

times n

which is 50.

so 2 plus 100 that's 102 divided by 2

that's 51.

51 times 50

is 2550.

so that is the sum

of all of the even numbers from 2 to 100

inclusive

try this one determine the sum of all

odd integers from 20 to 76

20 is even but the next number 21 is odd

and then 23 25 27

all of that are odd numbers up until 75

so a sub 1

is 21 in this problem

the last number a sub n

is 75

and we know the common difference is two

because the numbers are increased by two

what we need to calculate is the value

of n

once we could find n then we could find

the sum from

21 to 75.

so what is the value of n

so we need to use

the general formula for

an arithmetic sequence

so a sub n is 75 a sub 1 is 21

and the common difference is 2.

so let's subtract both sides by 21

75 minus 21 this is going to be 54.

dividing both sides by 2.

54 divided by 2 is 27. so we get 27 is n

minus 1

and then we're going to add 1 to both

sides

so n is 28

so a sub 28 is 75

75 is the 28th term in the sequence

so now we need to find the sum of the

first 28 terms

it's going to be a sub 1 the first term

plus the last term or the 28th term

which is 75

divided by 2

times the number of terms which is

28

21 plus 75 that's 96

divided by 2 that's 48 so 48 is the

average of the first and the last term

so 48 times 28

that's 1

344.

so that is the sum of the first 28 terms

you