Geometric Series and Geometric Sequences - Basic Introduction

Summary

This educational video script explores the concepts of geometric sequences and series, distinguishing them from arithmetic ones by their common ratio versus common difference. It explains how to calculate the nth term and the partial sum of a geometric series, highlighting the formulae involved. The script also discusses the arithmetic and geometric means, provides examples of writing equations between terms, and covers the sum of infinite geometric series, emphasizing the convergence criteria. Practice problems are included to reinforce the concepts.

Takeaways

• 📚 The difference between a geometric sequence and a series is that a geometric sequence is a list of numbers with a common ratio between terms, while a geometric series is the sum of the numbers in a geometric sequence.
• 🔢 In a geometric sequence, each term is found by multiplying the previous term by a common ratio, whereas in an arithmetic sequence, each term is found by adding a common difference to the previous term.
• 🧮 The formula to find the nth term of a geometric sequence or series is \( a_n = a_1 \times r^{(n-1)} \), where \( a_1 \) is the first term and \( r \) is the common ratio.
• 📈 The partial sum formula for a geometric series is \( S_n = a_1 \times \frac{1 - r^n}{1 - r} \), which calculates the sum of the first \( n \) terms.
• ⚠️ The sum of an infinite geometric series can only be calculated if the series converges, which occurs when the absolute value of the common ratio \( r \) is less than 1.
• 🔁 The sum of an infinite geometric series that converges is given by \( S_{\infty} = \frac{a_1}{1 - r} \), where \( a_1 \) is the first term and \( r \) is the common ratio.
• 🔢 The arithmetic mean (Ma) of two numbers in an arithmetic sequence is the average, while the geometric mean (Mg) of two numbers is the square root of their product.
• 📉 To relate terms within a geometric sequence, multiply the previous term by the common ratio raised to the power of the term's position difference.
• 📝 When identifying a sequence or series, look for a common ratio (geometric) or common difference (arithmetic), and determine if it is finite or infinite based on whether it has an end or continues indefinitely.
• 📊 Practice problems in the script illustrate how to calculate terms of a geometric sequence, write general formulas, and determine the type of sequence or series based on given patterns.
• 🤓 Understanding the properties of geometric sequences and series is essential for solving problems involving series convergence, term calculation, and sum determination.

Q & A

What is the main focus of the video?

The video focuses on geometric sequences and series, explaining the difference between them, how to identify them, and how to calculate various terms and sums within these sequences and series.

What distinguishes a geometric sequence from an arithmetic sequence?

A geometric sequence is distinguished by a common ratio between consecutive terms, whereas an arithmetic sequence has a common difference between terms. In a geometric sequence, each term is found by multiplying the previous term by the common ratio.

How is the common ratio of a geometric sequence calculated?

The common ratio is calculated by dividing any term in the sequence by the previous term. For example, if the sequence is 3, 6, 12, ..., the common ratio is 2, since 6 divided by 3 equals 2.

What is the formula to find the nth term of a geometric sequence or series?

The formula to find the nth term of a geometric sequence or series is given by \( a_n = a_1 \times r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

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

The sum of the first n terms of a geometric series is calculated using the formula \( S_n = a_1 \times \frac{1 - r^n}{1 - r} \), where \( S_n \) is the sum, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

What is an infinite geometric series and when does it converge?

An infinite geometric series is a series that continues indefinitely. It converges when the absolute value of the common ratio \( r \) is less than 1, meaning the terms get smaller and smaller, allowing the series to sum to a finite value.

How is the sum of an infinite geometric series calculated?

The sum of an infinite geometric series is calculated using the formula \( S_{\infty} = \frac{a_1}{1 - r} \), where \( a_1 \) is the first term and \( r \) is the common ratio, provided that the absolute value of \( r \) is less than 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 2. The geometric mean is the square root of the product of the two numbers, reflecting the middle term in a geometric sequence.

How can you determine if a sequence is arithmetic or geometric by looking at its terms?

A sequence is arithmetic if there is a common difference between consecutive terms, while it is geometric if there is a common ratio by which you multiply one term to get the next.

What is the relationship between the terms in a geometric sequence defined by a recursive formula?

In a geometric sequence defined by a recursive formula, each term is found by multiplying the previous term by the common ratio, represented as \( a_n = r \times a_{n-1} \).

Can you provide an example of how to write the first five terms of a geometric sequence given the first term and the common ratio?

Certainly. If the first term \( a_1 \) is 2 and the common ratio \( r \) is 3, the first five terms would be calculated as follows: \( a_1 = 2 \), \( a_2 = 2 \times 3 = 6 \), \( a_3 = 6 \times 3 = 18 \), \( a_4 = 18 \times 3 = 54 \), and \( a_5 = 54 \times 3 = 162 \).

Outlines

00:00

📚 Introduction to Geometric Sequences and Series

This paragraph introduces the topic of geometric sequences and series, distinguishing them from arithmetic sequences by the presence of a common ratio in geometric sequences versus a common difference in arithmetic sequences. It provides an example sequence and explains how to identify the common ratio. The paragraph also explains the difference between a sequence and a series and introduces the formula for finding the nth term of a geometric sequence or series, which is the first term multiplied by the common ratio raised to the power of (n-1). An example calculation is provided to illustrate the formula's application.

05:04

🔍 Understanding Geometric Series and Means

The second paragraph delves into geometric series, which are the sums of numbers in a geometric sequence, and provides the formula for calculating the partial sum of a geometric series. It contrasts finite and infinite geometric series, explaining that infinite series continue indefinitely. The paragraph also introduces the concepts of arithmetic mean (average of two numbers) and geometric mean (square root of the product of two numbers), using examples from arithmetic and geometric sequences to illustrate how to find these means.

10:05

🔢 Writing Equations and Summing Infinite Geometric Series

This paragraph discusses how to write equations between terms within a geometric sequence, emphasizing the role of the common ratio in relating terms. It explains that to move from one term to another, you multiply by the common ratio raised to the power of the difference in their positions. The paragraph also addresses how to calculate the sum of an infinite geometric series, noting that it converges and can be summed if the absolute value of the common ratio is less than one. Examples are provided to demonstrate these concepts.

15:07

📝 Practice Problems on Geometric Sequences

The fourth paragraph presents practice problems involving writing the first five terms of given geometric sequences, using the common ratio to find subsequent terms. It also includes writing a general formula for the nth term of a geometric sequence and calculating the value of a specific term, such as the eighth term. The paragraph guides through the process of identifying the common ratio and applying it to find terms in the sequence.

20:07

📉 Describing Patterns in Sequences and Series

In this paragraph, the task is to describe patterns in numbers as either arithmetic or geometric, finite or infinite, sequences or series. The explanation involves identifying whether there is a common difference or ratio and the presence of plus signs or commas to determine if it's a series or sequence. It also discusses whether the pattern continues indefinitely or has a clear end to classify it as finite or infinite.

25:09

🧮 Calculating Sums of Geometric Sequences

The sixth paragraph focuses on calculating the sum of the first ten terms of a geometric sequence and the sum of an infinite geometric series. It explains the formula for the sum of the first n terms of a geometric sequence and how to apply it, as well as the criteria for an infinite geometric series to converge (absolute value of the common ratio less than one). The paragraph provides step-by-step calculations for both scenarios.

30:10

🏁 Conclusion on Summing Infinite Geometric Series

The final paragraph wraps up the discussion on infinite geometric series, providing a formula for calculating their sum and emphasizing the condition for convergence (absolute value of the common ratio must be less than one). It includes a calculation example to find the sum of a specific infinite geometric series, demonstrating the process clearly.

Keywords

💡Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the video, the sequence 3, 6, 12, 24, 48 is given as an example, where the common ratio is 2. This concept is central to understanding the theme of the video, which is the mathematical properties and calculations related to sequences and series that follow a multiplicative pattern.

💡Geometric Series

A geometric series is the sum of the terms in a geometric sequence. The video explains that it is the cumulative total of numbers that follow a common ratio, such as the series 3 + 6 + 12 + 24 + 48. The concept is integral to the video's exploration of how to calculate the sum of a set number of terms in a geometric progression.

💡Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply one term to get the next. The video script uses the example of the sequence 3, 6, 12, etc., where each term is twice the previous one, thus having a common ratio of 2. The common ratio is essential for identifying a geometric sequence and for calculating further terms or the sum of the series.

💡Arithmetic Sequence

An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. The video contrasts this with a geometric sequence, using the sequence 5, 8, 11 as an example, where each term increases by a common difference of 3. Understanding the difference between arithmetic and geometric sequences is crucial for the video's discussion on sequences.

💡Partial Sum

The partial sum of a series refers to the sum of a certain number of terms from the beginning of the series. In the video, the formula for the partial sum of a geometric series is given as the first term times (1 - r^n) / (1 - r), where 'n' is the number of terms. The video uses this formula to calculate the sum of the first five terms of a geometric series, illustrating its application in finite series.

💡Infinite Geometric Series

An infinite geometric series is a series that continues indefinitely, with no end term. The video explains that such a series can be summed if the absolute value of the common ratio is less than one, using the formula a / (1 - r), where 'a' is the first term and 'r' is the common ratio. The video provides examples of both divergent and convergent infinite series.

💡Arithmetic Mean

The arithmetic mean, or average, of a set of numbers is found by summing all the numbers and then dividing by the count of numbers. In the context of the video, the arithmetic mean is used to find the middle term between two numbers in an arithmetic sequence, such as averaging 5 and 11 to get 8.

💡Geometric Mean

The geometric mean of a set of numbers is the nth root of the product of the numbers, where 'n' is the count of the numbers. In the video, the geometric mean is used to find the middle term between two numbers in a geometric sequence, such as finding the square root of 36 (3*12) to get 6, which is the middle term between 3 and 12.

💡Recursive Formula

A recursive formula defines each term of a sequence in relation to the previous term. In the video, the concept is used to describe how to find subsequent terms in a geometric sequence once the first term and the common ratio are known, such as a_n = a_(n-1) * r.

💡Convergence

In the context of series, convergence refers to the property of a series where the sum approaches a finite value. The video explains that an infinite geometric series converges if the absolute value of the common ratio is less than one, allowing for the calculation of a finite sum.

💡Divergence

Divergence in a series means that the sum does not approach a finite value but instead grows without bound. The video clarifies that an infinite geometric series diverges if the absolute value of the common ratio is greater than one, making it impossible to calculate a sum.

Highlights

Introduction to the difference between geometric sequences and series.

Example of a geometric sequence with a common ratio.

Explanation of how to distinguish between arithmetic and geometric sequences.

Formula for calculating the nth term of a geometric sequence or series.

Demonstration of calculating the fifth term of a geometric sequence.

Partial sum formula for the sum of the first n terms of a geometric series.

Calculation of the sum of the first five terms of a geometric series.

Identification of infinite geometric series and their properties.

Concept of arithmetic and geometric means and their differences.

Finding the arithmetic mean between terms in an arithmetic sequence.

Calculating the geometric mean between terms in a geometric sequence.

Writing equations between terms within a geometric sequence using common ratios.

Formula for calculating the sum of an infinite geometric series and its conditions for convergence.

Examples of calculating the sum of infinite geometric series with different common ratios.

Practice problems for writing the first five terms of given geometric sequences.

Writing a general formula for the nth term of a geometric sequence and calculating specific terms.

Describing patterns of numbers as arithmetic or geometric, finite or infinite, sequence or series.

Finding the sum of the first ten terms of a geometric sequence with a given common ratio.

Summation of an infinite geometric series with a specific common ratio and its convergence criteria.

Transcripts

in this video we're going to focus on

geometric sequences and series

so first let's discuss the difference

between

a geometric sequence

and

a geometric series

what do you think the difference is

here's an example of a geometric

sequence

the numbers 3

6

12

24

48 and so forth

a geometric sequence

is different from

an arithmetic sequence

such as

this one here

in that a geometric sequence has a

common ratio versus a common difference

if you take the second term and divide

it by the first term

six divided by three is two

you're going to get the common ratio

if you take the third term divided by

the second term you'll get the same

common ratio 12 divided by 6 is 2.

so that's the defining mark of a

geometric sequence

in an arithmetic sequence there's a

common difference

if you take the second term and subtract

it from the first term

eight minus five is three

if you take the third term subtracted by

the second

eleven minus eight is three

so that's how you could distinguish an

arithmetic sequence from a geometric

sequence an arithmetic sequence has a

common difference between terms

a geometric sequence has a common ratio

between terms

within an arithmetic sequence you're

dealing with addition and subtraction

for a geometric sequence you're dealing

with multiplication and division between

terms

so now that we know what a geometric

sequence is and how to distinguish it

from an arithmetic sequence what is the

geometric series

a geometric series is basically

the sum of the numbers in a geometric

sequence

so 3 plus 6 plus 12

plus 24

and so forth

would be a geometric series

this is the first term this is the

second term this is the third

and so forth

now the formula you need to calculate

the f term of a geometric sequence

or series

it's

the first term a sub 1

times the common ratio r raised to the n

minus 1. so for instance

let's just make a note that r is equal

to 2.

let's say we want to find the value of

the fifth term we know the fifth term is

48 but let's go ahead and calculate it

so you can see how this formula works

so the first term is three

r

the common ratio is two

and n is the subscript here we're

looking for the fifth term so

n is five

so five minus one is four

two to the fourth power if you multiply

two four times two times two times two

times two that's 16.

16 times 3 is 48

so that's the function of this formula

it gives you the value of the f term so

you can find the value of the eighth

term the 20th term and so forth

the next equation you need to be

familiar with

first let's get rid of this

the next equation is the partial sum

formula

the partial sum of a geometric series

is the first term times 1 minus r raised

to the n

over

1 minus r

so let's say that we want to find the

sum of the first five terms

this is going to be 3

plus 6

plus 12

plus 24.

plus 48

go ahead and plug that into a calculator

so for the first five terms i got the

partial sum as being 93.

now let's confirm that with this

equation

so let's calculate s sub 5

the first term is 3

times 1 minus r r is 2 n is 5

divided by 1 minus r so that's 1 minus

2.

2 to the fifth power

that's 32

1 minus 2 is negative 1.

now 1 minus 32 is negative 31.

3 times negative 31 that's negative 93

but divided by negative 1 that becomes

positive 93

so we get the same answer

so anytime you need to find

the sum

of a finite series

you could use this formula

so

this series here is finite

we're looking for the sum of the first

five terms there's beginning and there's

an end

this series here

is

not finite it's an infinite geometric

series

the reason being is because of the dot

dot that we see here it goes on forever

it doesn't stop at the fifth term it

keeps on going to infinity so it's an

infinite geometric series

this

is an infinite

geometric sequence

it's a sequence that goes on from

forever and it's geometric so make sure

you can identify

if a sequence is arithmetic geometric is

it finite infinite is it a sequence or a

series

now the next thing we need to talk about

is

the arithmetic mean and the geometric

mean

let's call the arithmetic mean m a

the arithmetic mean is simply the

average of two numbers

the geometric mean let's call it mg

is the square root of the product of two

numbers

so let's go back to

the arithmetic sequence that we had here

if we wanted to find

the arithmetic mean between the first

and the third term it will give us the

middle number the second term

if you average 5 and 11 and divide by 2

using this formula

you're going to get 16 over 2

which is eight

so thus when you find the arithmetic

mean of the first term

and the third term

you're going to get the second term

because the average of one and three is

two

now

let's find the arithmetic mean between

the first and the fifth term

this will give us

the middle term 11.

so if we were to add up a1

and a5

and then divided by 2 if we were to get

the average

we would get a3 the average of 1 and 5

is three

so let's add five and seventeen and then

divide by two

five plus seventeen is twenty two

twenty two divided by two is eleven

so that's the concept of the arithmetic

mean whenever you take

the arithmetic mean of two numbers

within an arithmetic sequence you get

the middle term of that of those two

numbers that you selected

now the same is true for a geometric

sequence

if we were to find the geometric mean

between 3 and 12

we would get the middle number 6.

if we wanted to find the geometric mean

between 3 and 48 we would get the middle

number 12.

so let's confirm that

let's find the geometric mean

between a1 and a3

so the first term is 3

the third term is 12.

3 times 12

is 36

the square root of 36 is 6.

so we get the middle number

now let's find the geometric mean

between the first term and the fifth

term

so we should get 12 as an answer

so the average of one and five one plus

five is six divided by two is three so

we should get a sub three

the first term is 3 the last term or the

5th term is 48.

now what's 3 times 48

if you're not sure what you could do is

break it up into

smaller numbers

48 is three times sixteen

three times three is nine so you have

the square root of nine times the square

root of sixteen

the square root of nine is three the

square root of sixteen is four three

times four is twelve

so the geometric mean

of 3 and 48

is the middle number in the geometric

sequence which is 12.

now sometimes you need to be able to

write equations between

terms within a geometric sequence

for instance if you want to relate the

second equation

to the first equation you need to

multiply by r

i mean the second term to the first term

if you want to relate the fifth term to

the second term you need to multiply by

r cubed

to go from the second term to the fifth

term you need to multiply it by r three

times

if you multiply six by r you're going to

get 12. if you multiply 12 by r you get

24.

24 by r you get 48.

so to go from the second term to the

fifth term you need to multiply by r

cubed

and the reason why it's cube is because

the difference between five and two is

three

and you could check that so if you take

the second term which is six multiply it

by two to the third that's six times

eight which is 48 and that gives you the

fifth term

so if i want to relate the ninth term

to the fourth term

how many r values do i got to multiply

the fourth term to get to the ninth term

nine minus four is five

so i gotta multiply the fourth term by r

to the fifth power to get the ninth term

so make sure you know how to write those

formulas

so we've discussed calculating the sum

of a finite series

just review if you want to calculate the

sum

of a finite series one that has a

beginning and an end

you would use this formula

now what about the sum

of an infinite series

how can we find that

what's the formula

that we need to calculate s to infinity

it's basically this same formula but

without that part it's a one over one

minus r

so here's two examples

of an infinite geometric series this is

one of them

and this one is going to be another one

eight

four

two

one

one half

and so forth

we can't calculate the sum of both

infinite geometric series

for this one r is two

so r

or rather the absolute value of r is

greater than one

when that happens

the geometric series diverges

which means you can't calculate the sum

because it doesn't

it doesn't converge to a specific value

if you keep adding these numbers

it's not going to converge to a value

it's going to get bigger and bigger and

bigger

so the series diverges

if you try to calculate it let's say you

plugged in 1 for a1

and

2 for r it's not going to work

you get 3 over negative 1 which is

negative 3 and clearly that's not the

sum of this series

the fact that

you get a negative sum from positive

numbers tells you something is wrong

so this formula

doesn't work if the series diverges it

only works if the series converges and

that happens

when the absolute value of r is less

than one

if we focus on this particular infinite

geometric series

notice the value of r

if we take the second term divided by

the first term

four over eight is one half

if we take the third term divided by the

second term

two over four reduces to one half

so that's the value of r

so for that particular series we could

say that

the absolute value of r which is one

half

that's less than one

therefore

the series

converges

which means we can calculate a sum it

has a finite sum even though the numbers

get smaller and smaller and smaller

now let's calculate the sum

so the sum

of an infinite number of terms of this

geometric series is going to be the

first term a sub 1 which is 8

over 1 minus r

where r is a half

one minus one half is one half

so multiplying the top and bottom by two

we get 16 on top

these two will cancel we get one

so the sum of this infinite geometric

series that converges is 16.

so that's how you can calculate the sum

of an infinite geometric series

the series must converge and for that to

happen the absolute value of r has to be

less than one

if it's greater than one the series will

diverge and you won't be able to

calculate the sum

now let's work on some practice problems

write the first five terms of each

geometric sequence shown below

so let's start with the first one

the first term is two

to find the next term we need to

multiply

the first term by the common ratio the

second term is equal to the first term

times the common ratio

so 2 times 3 is 6.

and then to get the third term we just

got to multiply the second term by the

common ratio

6 times 3 is 18

18 times 3 is 54

and then 54 times 3 that's 162.

so that's the answer for

number one

let's move on to number two

the first term is 80.

the common ratio is one-half

so we're going to multiply 80 by a half

half of 80 is 40

half of 40 is 20

half of 20 is 10

half of 10 is five

so those are the first five terms for

the second geometric sequence

now let's move on to number three

so the first term is six

to find the next term we need to

multiply six by negative two

so this is going to be negative twelve

negative twelve times negative two is

positive 24

and then it's just going to alternate

so whenever you see a sequence a

geometric sequence with

alternating signs

then you know that the common ratio must

be negative

number two

write the first five terms of the

geometric sequence defined by the

recursive formula shown below

so we're given the first term

when n is 2 we have that the second term

is equal to negative 4

times the first term

and we know that the second term

is the first term times r

so therefore r

the common ratio must be negative 4.

so anytime you need to write a recursive

formula of a geometric sequence it's

going to be a sub n is equal to r

times

the previous term a sub n minus 1.

the next term is always the previous

term times the common ratio

so the common ratio

is this number negative four

so once we have the first term in the

common ratio we can easily write out the

sequence so the first term is negative

three

the second term will be negative three

times negative four

which is twelve

the third term

will be

net twelve times negative four which is

negative forty eight

the fourth term is negative 48 times

negative 4

which is 192

and then the fifth term

192 times negative 4

is negative

768

so that's how we can write the first

five terms of the geometric sequence

defined by recursive formula it's by

realizing that this number is the common

ratio

write a general formula that gives the f

term of each geometric sequence

and then calculate the value of the

eighth term

of each of those geometric sequences

so let's start with number one

so we have the number 6

24

96

384 and so forth

the first thing we need to do is

calculate the common ratio

so let's divide the second term by the

first term

dividing 24 by six we get four

now just to confirm that

this is indeed a geometric sequence

let's take the third term and divide it

by this the second term not the first

one

so 96 divided by 24

and that is also equal to 4.

so we have a geometric sequence here

in order to write the formula

all we need is the value of the first

term and the common

ratio so we could use this equation the

f term is going to be equal to the first

term times r

raised to the n minus one

the first term being six

r is four

so we can write it as a sub n is equal

to 6

times 4 raised to the n minus 1.

so this is the answer for part a for the

first

sequence

now let's move on to part b let's

calculate the value

of the eighth term

so we just got to plug in 8 into n

so it's 6

times 4 raised to the eight minus one

eight minus one is seven

four raised to the seventh power is

sixteen thousand three hundred eighty

four

times six

this gives us ninety eight thousand

three hundred and four

so that is the value of the eighth term

and you could confirm it

if you keep

multiplying these numbers by 4 you're

going to get it

384

times 4

that's

15 36 that's the fifth term

if you times it by 4 again

you get sixty one forty four

times four you get

twenty four five seven six

and then times four gives you this

number

now let's move on to number two

so we have the sequence

5

negative 15 45

negative 135 and so forth

so the first term is five

the common ratio

which can be calculated by taking the

second term divided by the first term

that's negative 15 divided by five

that's negative three

r is also equal to the third term

divided by the second term

so that's 45

over

negative 15 which is negative three

so the value of the first term is five

r

is negative three

so now let's go ahead and write a

general formula that gives us the nth

term

so a sub n is going to be a sub 1 times

r raised to the n minus 1.

the first term is 5

r is negative three

so this right here is the answer

for part a

now part b calculate the value of the

eighth term

so let's replace n with eight

negative three raised to the seventh

power

that's negative two thousand one hundred

and eighty-seven

multiplying that by five

this gives us negative ten thousand

nine hundred and thirty-five

so that's the final answer for part b

number four

describe each pattern of numbers as

arithmetic or geometric finite or

infinite sequence or series

let's look at the first one

do we have a common difference or common

ratio

going from 4 to 8

we increase by four

from eight to twelve that's an increase

by four

and twelve to sixteen

so we're constantly adding four we're

not multiplying by four

so therefore

we have a common difference and not a

common ratio

so because we have a common difference

this is arithmetic

not geometric

we're dealing with addition rather than

multiplication

now is this a sequence or series

we're not added numbers so

we have a sequence

if you see a comma between the numbers

it's going to be a sequence if you see a

plus you're dealing with a series

now

is this sequence finite or infinite

it has a beginning and it has an end

we don't have

dots that indicate that it goes on

forever so this is finite so we have a

finite arithmetic sequence

for number one now let's move on to

number two

going from 90 to 30 that's a difference

of negative 60. going from 30 to 10

that's a difference of negative 20.

so we don't have a common difference

here

if we divide the second term by the

first term

this reduces to one-third

if we divide the third term by the

second term that's also

one-third so what we have here

is a common ratio

rather than a common difference

so the pattern of numbers is geometric

not arithmetic

we're multiplying by 1 3

to get the second term from the first

term

now are we dealing with a sequence or

series

so we don't have a plus sign between a

number so we have a comma

so we're dealing with a sequence

and this sequence has no end it goes on

forever

so what we have here is an infinite

geometric sequence

now for number three

we could see that we have a common ratio

of two

five times two is ten ten times two is

twenty twenty times two is forty

so this is geometric

now there's a plus between the numbers

so this is going to be a series not a

sequence

and this sequence i mean the series

rather comes to an end the last number

is 80. so it's not infinite but it's

finite

so we have a finite geometric series

for the last one we can see that we have

a common difference of negative four

fifty minus four

is forty six forty six minus four is 42

so this is going to be arithmetic

we have a plus between a number so it's

a series

and it goes on forever

so it's infinite

so we have an infinite arithmetic series

number five

find the sum of the first ten terms of

the geometric sequence shown below

so the first term is 7

the common ratio

negative 14 divided by 7

that's negative 2.

28 divided by negative 14 is also

negative 2.

so now that we know the first term in

the common ratio we can calculate the

sum

using this formula

so it's the first term times one minus r

raised to the n

over one minus r

so the sum of the first 10 terms

it's going to be the first term 7

times 1 minus

negative 2

now it's raised to the n power n is 10

and then divided by 1

minus r

negative two

raised to the tenth power

that's positive one thousand

twenty four

and here we have one minus negative two

which is one plus two and that's three

one minus

ten twenty four

that's negative one thousand

twenty three

seven times negative ten twenty three

divided by three

that's negative two thousand three

hundred and eighty seven

so that's the final answer

try this problem find the sum of the

infinite geometric series

so we have the numbers 270

90

30

10

and so forth

so we can see that the first term a sub

1

that's 270.

the common ratio if we divide 90 by 70

what does that simplify to

i mean 90 by 270.

well we could cancel a 0 so we get 9

over 27

9 is 3 times 3 27 is 9 times three

so that becomes true over nine

three we can write that as three times

one nine is three times three this is

one third

so that's going to be the common ratio

and of course if you divide

30 by 90 you also get one-third

so now that we have the first term and

the common ratio

we can now calculate

the sum of the infinite geometric series

using this formula it's going to be a

sub 1 over 1 minus r

by the way

this particular infinite geometric

series does a converge converger diverge

the absolute value of r

is less than one it's one third which is

about 0.333 or 0.3 repeating

so because it's less than 1

the infinite geometric series

converges

we can calculate the sum the sum is

finite

so let's go ahead and calculate that sum

the first term is 270

r

is one third

so what's one minus one third

one if you multiply it by three over

three

you get three over three minus one over

three which is

two over three

now what i'm going to do is i'm going to

multiply the top and bottom by three

these threes will cancel

so it's going to be 270

times three divided by two

two seventy divided by two is one thirty

five one thirty five times three

and that's going to be four oh five

so that is the sum

of this infinite geometric series

you