Grade 10 Math Q1 Ep5: Finding the Sum of the Terms of a Given Arithmetic Sequence

Summary

In today's episode of 'adeptv', Sir Jason Flores guides viewers through the mathematical concepts of arithmetic sequences. The lesson focuses on calculating the sum of the first 'n' terms of such sequences, introducing formulas to simplify the process. Examples are provided to demonstrate how to use these formulas when both the first and last terms are known, or when only the first term and the common difference are given. The episode concludes with a real-world application, showing how these mathematical skills can be applied to saving money, as illustrated by a story about Jane saving for shoes.

Takeaways

• 📘 The lesson focuses on teaching the formula to find the sum of the first n terms of an arithmetic sequence, which is crucial for solving word problems involving series.
• 🔢 The formula for the sum of the first n terms when the first and last terms are known is S_n = (n/2)(a_1 + a_n).
• 🔄 When the last term is not given, an alternative formula is used: S_n = (n/2)(2a_1 + (n-1)d), where d is the common difference.
• 📝 An example is provided to demonstrate the calculation of the sum of the first 20 natural numbers, which equals 210.
• 💡 The lesson emphasizes the practicality of using formulas over manual addition for sequences with many terms, highlighting efficiency in computation.
• 📊 A step-by-step approach is shown for calculating the sum of terms in sequences, such as 5, 10, 15, ... up to 50, which sums to 275.
• 👟 A real-world application is presented where a student, Jane, saves money weekly, and the formula is used to calculate her total savings after 43 weeks.
• 🌟 The lesson concludes with a motivational message encouraging continuous practice and highlighting the relevance of math in daily life.
• 🎓 Sir Jason Flores, the presenter, aims to make learning math fun and easy, emphasizing the importance of logical reasoning and critical thinking skills.

Q & A

What is the main topic discussed in this episode?

The episode focuses on finding the sum of the first n terms of an arithmetic sequence and solving word problems involving arithmetic series.

What formula is used to find the sum of the first n terms when both the first and last terms are given?

The formula used is S(n) = n / 2 * (a1 + an), where S(n) is the sum, n is the number of terms, a1 is the first term, and an is the last term.

How do you find the sum of the first n terms when the last term is not given?

When the last term is not given, the formula is S(n) = n / 2 * [2a1 + (n - 1)d], where d is the common difference between terms.

In the example, what is the sum of the first 20 natural numbers?

The sum of the first 20 natural numbers is 210.

How is the sum of the sequence 5, 10, 15, 20, up to 50 calculated?

By listing all the terms and adding them together, the sum is calculated as 275.

What is the result when calculating the sum of the first 16 terms of the sequence 8, 11, 14, 17, 20, etc.?

The sum of the first 16 terms of this sequence is 488.

What is the formula used to solve the word problem about Jane saving money, and what is the final result?

The formula used is S(n) = n / 2 * [2a1 + (n - 1)d], and after 43 deposits, Jane saves a total of 3,827 pesos.

What is the difference in the arithmetic sequence 1, 3, 5, 7, and so on?

The common difference in this sequence is 2.

How is the sum of the sequence -3, -1, 1, 3, etc., calculated?

Using the formula for when the last term is not given, the sum of the first 13 terms is 117.

What message does the episode conclude with regarding math?

The episode encourages viewers to keep practicing math, emphasizing that math is part of daily life and can be fun and easy.

Outlines

00:00

📘 Introduction to Arithmetic Sequences

Sir Jason Flores introduces the lesson on arithmetic sequences, focusing on developing logical reasoning and critical thinking skills. The lesson aims to teach how to find the sum of the first n terms of an arithmetic sequence and solve related word problems. The importance of having learning modules, pens, and paper ready is emphasized. The previous lesson's discussion on finding the nth term of an arithmetic sequence using the formula a_n = a_1 + d(n-1) is recalled, and the new focus on the sum of the first n terms is introduced. Examples of sequences are given, and the impracticality of manual addition for long sequences is highlighted, leading to the introduction of a formula for faster computation.

05:02

🔢 Sum of Arithmetic Sequences Formulas

The video script explains two formulas for finding the sum of the first n terms of an arithmetic sequence. The first formula, S_n = (n/2)(a_1 + a_n), is used when the first and last terms are known. The second formula, S_n = (n/2)(2a_1 + (n-1)d), is used when the last term is not given. Examples are provided to illustrate the application of these formulas. The first example calculates the sum of the first 20 natural numbers, and the second example finds the sum of the first 16 terms of a sequence starting with 8 and having a common difference of 3. The process of substituting values into the formulas and performing the calculations is detailed.

10:05

📈 Practical Examples of Sum Calculations

This section presents practical examples to apply the learned formulas for calculating the sum of arithmetic sequences. The first example involves finding the sum of the first 30 terms of a sequence starting with 1 and having a common difference of 2. The formula S_n = (n/2)(2a_1 + (n-1)d) is used, and the calculation leads to a sum of 900. The second example calculates the sum of the first 10,000 terms of a sequence starting with 1 and ending with 10,000, using the formula S_n = (n/2)(a_1 + a_n), resulting in a sum of 50,005,000. These examples demonstrate the efficiency of the formulas in handling large sequences.

15:09

💼 Applying Arithmetic Sequences to Real-life Problems

The script transitions into applying arithmetic sequences to real-life scenarios with a word problem about Jane saving money for shoes. The problem describes an arithmetic sequence where Jane saves a weekly increasing amount starting from 5 pesos. The goal is to find out how much she saves after 43 weeks. The formula S_n = (n/2)(2a_1 + (n-1)d) is applied with the given values, leading to a total savings of 3,827 pesos. This example ties mathematical concepts to a relatable situation, showing how arithmetic sequences can be used in everyday life.

20:13

🎓 Summary and Conclusion

The final part of the script summarizes the key formulas for finding the sum of an arithmetic sequence: S_n = (n/2)(a_1 + a_n) when the first and last terms are given, and S_n = (n/2)(2a_1 + (n-1)d) when the last term is not given. A word problem involving Jane's savings is used to reinforce the learning, emphasizing the practical application of these formulas. The lesson concludes with an encouragement to practice and apply mathematical concepts in daily life, highlighting the fun and ease of learning math, and looking forward to the next episode.

Keywords

💡Arithmetic Sequence

An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is known as the common difference. In the video, arithmetic sequences are used to teach how to find the sum of the first n terms using specific formulas. For example, the sequence '1, 2, 3, ..., 100' is an arithmetic sequence with a common difference of 1.

💡Sum of the First n Terms

The sum of the first n terms refers to the total when the first n numbers in a sequence are added together. The video introduces formulas to calculate this sum efficiently, especially useful for sequences with many terms. For instance, the sum of the first 20 natural numbers is calculated as 210 in the script.

💡Common Difference (d)

The common difference in an arithmetic sequence is the constant amount by which each term increases from the previous one. It is a fundamental concept used in the formulas for finding the nth term and the sum of the first n terms. In the script, sequences like '5, 10, 15, 20, ...' have a common difference of 5.

💡nth Term (a_n)

The nth term of a sequence refers to the number that is in the nth position of that sequence. The video script discusses a formula to find the nth term when the first term and the common difference are known: a_n = a_1 + (n - 1)d. This is crucial for understanding how to find any term in an arithmetic sequence.

💡Logical Reasoning

Logical reasoning is the process of thinking that uses logic to arrive at conclusions. The video aims to develop this skill by teaching viewers how to apply mathematical formulas and concepts to solve problems, such as finding the sum of terms in an arithmetic sequence.

💡Critical Thinking Skills

Critical thinking skills involve analyzing and evaluating information to form judgments. The video encourages the development of these skills by presenting mathematical problems that require viewers to apply formulas and think critically to solve them.

💡Formula for Sum of an Arithmetic Sequence

The video provides formulas to find the sum of an arithmetic sequence. When the first and last terms are known, the formula S_n = n/2 * (a_1 + a_n) is used. If the last term is not known, the formula S_n = n/2 * [2 * a_1 + (n - 1) * d] is applied. These formulas are demonstrated through various examples in the script.

💡Word Problems

Word problems are practical, real-world scenarios presented in narrative form that require mathematical solutions. The video uses word problems involving arithmetic sequences to illustrate how mathematical concepts are applied in everyday contexts, such as saving money for a pair of shoes as described in the script.

💡Learning Module

A learning module refers to a structured set of learning materials designed to teach specific concepts or skills. The video is part of a learning module aimed at helping viewers develop their logical reasoning and critical thinking skills through the study of arithmetic sequences.

💡Sequence

A sequence is an ordered list of numbers or objects. In the context of the video, sequences are used to demonstrate arithmetic progressions and to teach how to calculate the sum of their terms. The video script discusses sequences like '1, 2, 3, ...' and '5, 10, 15, ...'.

Highlights

Introduction to the episode focusing on arithmetic sequences and their sum.

Explanation of how to find the sum of the first n terms of an arithmetic sequence.

Practical example of manually adding terms of a sequence to find the sum.

Discussion on the impracticality of manual addition for sequences with many terms.

Presentation of the formula for the sum of an arithmetic sequence when the first and last terms are given.

Presentation of the formula for the sum when the last term is not given.

Example calculation of the sum of the first 20 natural numbers.

Example calculation of the sum of a sequence with a given first and last term.

Example calculation of the sum of a sequence without a given last term.

Exercise to find the sum of the first 30 terms of an arithmetic sequence.

Exercise to find the sum of a sequence up to ten thousand terms.

Exercise to find the sum of the first 13 terms of a sequence with a negative first term.

Word problem involving saving money in an arithmetic sequence pattern.

Solution to the word problem calculating total savings after 43 deposits.

Conclusion and encouragement for continued learning and practice in mathematics.

Transcripts

[Music]

hi

good day welcome in today's episode of

adeptv

i am sir jason flores also your math

buddy and i will be here to help you in

developing your logical reasoning

and critical thinking skills

is yourself learning module ready

what about your pen and paper

great let's begin a fun and exciting

lesson

for this lesson you are expected to

first find the sum of the first

n terms of an arithmetic sequence

and second solve word problems

involving arithmetic series

in the previous episode it was discussed

that

to find the nth term of an arithmetic

sequence

the formula a sub n is equal to

a sub 1 plus d times

n minus 1 can be used

in this episode we will discuss how to

find

the sum of the first n terms of a given

arithmetic sequence

for example how do we compute the sum

of all the terms of each of the

following sequences

letter a 1 2

3 up to 100

and letter b 5 10

15 20 up to 50.

adding manually the terms of a sequence

is manageable

when there are only few terms in the

sequence

however if the sequence involves

numerous terms

then it is no longer practical to be

adding the terms

manually

it is a tedious work to do thus

this episode will present to you a

formula

that will make the computation faster

and easier to let you experience

getting the sum of the terms in a

sequence manually

let's do the following number one

find the sum of the first 20

natural numbers by listing

all the natural numbers from 1 to 20 and

adding them we have 1

plus 2 plus 3 plus 4

up to 20 that is equal to 210

thus the sum of the first 20

natural numbers is 210

number two find the sum of

all the terms of the sequence 5

10 15 20

up to 50. by listing all the terms of

the sequence

and adding them we have 5

plus 10 plus 15 plus

20 up to 50 you will get

275 thus

the sum of the terms of the sequence is

275

what if you're asked to find the sum of

the terms of the sequence

one two three

up to ten thousand there are ten

thousand terms to be added one by one

to get their sum right

in doing this kind of solution it is

very

challenging especially if you're dealing

with a sequence

that has many terms

in getting the sum of the terms of an

arithmetic sequence

we will be using any of the following

formula

s of n is equal to n

divided by 2 times a sub 1

plus a sub n if

the first and last term

are given again if the first

and last term are given we will use the

formula

s of n is equal to n divided by

2 times a sub 1 plus a sub

n where s sub n is

the sum of the first n terms

n corresponds to the nth position

a sub 1 is the first term

of the sequence and a sub n is

the last term

on the other hand if the last term is

not given

we will use the formula sub n

is equal to n divided by 2

times the quantity 2 times a sub 1

plus n minus 1 times d

again if the last term

is not given we will use the formula

s of n is equal to n divided by 2

times the quantity 2 times a sub 1

plus n minus 1 times d

where s sub n is the sum of the first

n terms a sub 1 is the first

term of the sequence and d

is the common difference

let's take a look at these examples

find the sum of the first 20

natural numbers from this sequence

we can see that our a sub 1 is equal to

1

our a sub n is equal to 20

our n is equal to 20 and we are looking

for s sub n since the first

and last terms are given we will use

the formula sub n

is equal to n divided by 2

times a sub 1 plus a sub n

substituting the given values in the

formula we will have

s sub 20

is equal to our n is 20

divided by 2 times

our a sub 1 is 1

plus our a sub n is equal to

20. next we will have

s sub 20 is equal to

20 divided by 2 will give us 10

and one plus 20 will give us

21. our s sub 20 will be the product of

10 and 21. we will get

210.

therefore the sum of the first

20 natural numbers is 210

now let's go to the next example

find the sum of the first 16 terms

of the arithmetic sequence 8

11 14 17

20 and so on

notice that in this sequence the given

are a sub 1 which is equal to 8 our n

is equal to 16. the difference

is 3 and we are looking for the sum

of the first 16 terms notice

that the last term is not given

so we will use the formula

sub n

is equal to n

divided by 2 times the quantity 2

times a sub 1 plus

n minus 1 times d

by substituting the given values in the

formula

we will have s

sub 16 is equal to

our n is 16

divided by two times the quantity two

our a sub one is eight

plus our n is

sixteen minus one

times the difference which is three

next we will have s sub 16

is equal to 16 divided by 2 is 8

times the quantity 2 times 8 will give

us

16 plus 16 minus 1

is 15 and the difference of

3.

next we will have s sub 16

is equal to 8 times the quantity sixteen

plus fifteen times three

will give us forty five

next we'll have s sub 16

is equal to 8 times the quantity of 16

plus 45 will give us

61.

our s of 16 is equal to the product of 8

and 61 that is equal to

488

thus the sum of the first 16 terms of

the series

is 488

great now let's move forward to the next

example

after knowing all the needed concepts in

finding the sum of an arithmetic

sequence

let's try to answer the following

exercises

find the sum of each arithmetic series

the first one find the sum

of the first 30 terms of the arithmetic

sequence

1 plus 3 plus

5 and so on

notice that in this problem our given a

sub one is equal to one

our n is thirty

our difference is 2 as you can see

by adding the difference to the

preceding term

to get the next term

and we are looking for s sub 30.

with this problem the last term

is not given so we will use

the formula s of n is equal to

n divided by 2 times the quantity

2 times a sub 1 plus

n minus 1 times d

substituting the values in the formula

we will get

s sub 30 is equal to

our n again is 30 divided by 2

times the quantity 2 times a sub 1 which

is 1

plus our n again is 30

minus 1 times the difference

2. next we have s sub 30

is equal to 30 divided by 2 will give us

15.

times the quantity 2 times 1 is 2

plus 30 minus 1 will give us

29 times the difference

2. moving forward we will have

s sub 30 is equal to 15

plus or the quantity 2

plus 29 times 2 is equal to

58 s sub 30

is equal to 15 times the quantity

of 2 plus 58 that's

60. our s

of 30 therefore is the product of

60 and 15 you will get

900.

thus the sum of the first 30 terms of

the sequence

is equal to 900.

excellent let's boost up your learning

with another example

find the sum of the sequence one

two three four

up to ten thousand

from the sequence we can notice that the

given

a sub one or first term is equal to one

our n is equal to ten thousand

our a sub n is equal to ten thousand

and we are looking for s sub n

also notice that the first

and the last term are given

so we will use the formula s sub n

is equal to n divided by 2

times a sub 1 plus a sub n

substitute the values in the given

formula we will have

s sub ten thousand

is equal to our n again is ten thousand

divided by two times

the first term which is one plus

our a sub n which is 10

000.

next we have s sub 10 000

is equal to 10 thousand divided by two

will give us five thousand

times the sum of one and ten thousand

you will have ten thousand one

our s sub 10 000 now is the product

of five thousand and ten thousand one

it will give us

this term therefore the

s sub 10 000 of the sequence

is 50 million 5

000. oh so

awesome fasten your seat belts as we

proceed

to the next exercise

find the sum of the first 13 terms

of the sequence negative 3

negative 1 1 3

and so on notice

that from the sequence our given a sub 1

or the first term

is negative 3. our n

is 13. the difference

is 2 that is by adding the common

difference

to the preceding term to get the next

term

and we are looking for s sub 13.

since the last term is not given

we will use the formula s of

n is equal to n

divided by 2 times

the quantity 2 times a sub 1 plus

n minus 1 times the difference

again substitute the values in the

formula

we will have s sub 13

is equal to n which is 13

divided by 2 times the quantity 2

times a sub 1 that is negative 3

plus our n which is

13 minus 1 times the difference

which is 2. next we have s

sub 13 is equal to 13 divided by 2

will give us correct 6.5 times the

quantity 2

times negative 3 this will be equal to

negative 6 plus

13 minus 1 will give us 12.

and times the difference which is 2.

moving on we have s sub 13

is equal to 6.5

times the quantity negative 6.

plus the product of 12 and 2

will give us 24.

next we have s sub 13

is equal to 6

point five times

get the sum of negative six and 24

will give us positive

eighteen that's correct

then our s sub 13 now will be equal to

the product of 6.5 and 18.

we will have 117.

so the sum of the 13 terms of this

sequence is equal to 117

how's that oh so amazing

well done congratulations

dear students on this part let us see

what you have learned from today's

episode

by answering the following questions

to find the sum of the terms of an

arithmetic sequence

what formula can be used if the first

term and last term are given

if the first term and last term are

given we use the formula

s sub n is equal to n divided by 2

times a sub 1 plus a sub n

what about if the last term is not given

what formula can we use

if the last term is not given we use the

formula

s sub n is equal to n divided by

2 times the quantity 2 times a sub 1

plus n minus 1 times d

now let's apply what you have learned by

solving

the given word problem

jane was saving for a pair of shoes

from her weekly allowance she was able

to save

five pesos on the first week

nine pesos on the second

13 pesos on the third week and so on

if she continued saving in this pattern

and made 43 deposits

how much did jane save

let's analyze and solve the problem

together

from the given problem we can see that

our first

term or a sub 1 is equal to 5 pesos

our n is equal to 43

the difference is equal to 4

and we're looking for s sub n

since the last term is not given

we will use the formula s sub n

is equal to n divided by 2 times

the quantity 2 times a sub 1 plus

n minus 1 times d

substituting the values in the formula

we will have

s sub 43 is equal to 43 divided by 2

times the quantity 2 times 5 plus

43 minus 1 times 4.

s sub 43 is equal to 43 divided by 2

is equal to 21.5

times the quantity 2 times 5 is 10

plus 43 minus 1 is 42

times 4 s of 43 is equal to 21.5

times the quantity 10 plus the product

of 42 and 4 that is 168.

s of 43 is equal to 21.5

times the sum of 10 and 168

that's one hundred seventy eight s sub

forty three

is the product of twenty one point five

and one hundred seventy eight that is

equal to three

thousand eight hundred twenty seven

thus after 43 deposits

jane can save 3827

pesos like

jane you too can save

start now your five pesos

can make a difference

oh so awesome right

i hope you learned a lot keep practicing

because

math is always part of our daily lives

[Music]

and that concludes our lesson for today

see you again on the next episode

and this has been sir jason flores also

bear in mind that learning math will

always be fun and easy

be also be awesome

only here on dappa tv

[Music]

you