Arithmetic Series - Sum of the Terms of Arithmetic Sequence

Summary

In this video, Teacher Turgon explains the concept of arithmetic series, focusing on two key formulas for calculating the sum of terms in an arithmetic sequence. The first formula is used when both the first and last terms are known, while the second formula is applied when the first term and common difference are given. He solves two example problems: one calculating the sum of the first 20 terms and another for the first 40 terms of different arithmetic sequences. The video offers a clear, step-by-step explanation to help viewers understand and apply these formulas.

Takeaways

• 📚 The video discusses arithmetic series and how to find the partial sum of a given number of terms in an arithmetic sequence.
• 📐 Two formulas for calculating the sum of an arithmetic series are introduced: one that uses the first and last term, and another that uses the first term and common difference.
• 🧮 Formula 1: Sₙ = n * (a₁ + aₙ) / 2, where n is the number of terms, a₁ is the first term, and aₙ is the last term.
• 📏 Formula 2: Sₙ = n/2 * (2a₁ + (n-1) * d), where d is the common difference, n is the number of terms, and a₁ is the first term.
• ✍️ Example 1: The sum of the first 20 terms in a sequence with a₁ = 5 and aₙ = 62 is calculated using Formula 1, resulting in Sₙ = 670.
• 📝 Simplification Tip: Instead of directly multiplying large numbers, simplify fractions to make calculations easier.
• 🔢 Example 2: The sum of the first 40 terms in an arithmetic series with a₁ = 2 and a common difference of 3 is calculated using Formula 2, resulting in Sₙ = 2420.
• 🧑‍🏫 A common difference is found by subtracting consecutive terms in the sequence. In Example 2, the difference is 3 (e.g., 5 - 2 = 3).
• 📊 Formula selection depends on whether the last term of the series is provided or if only the common difference and first term are given.
• 🎥 The video encourages viewers to like, subscribe, and hit the notification bell for updates.

Q & A

What is an arithmetic series?

An arithmetic series is the sum of a specific number of terms in an arithmetic sequence, where each term after the first is obtained by adding a constant value, called the common difference.

What does S sub n represent in the formulas?

S sub n represents the sum of the first 'n' terms in an arithmetic series.

What is the formula for the sum of an arithmetic series when the first and last terms are known?

The formula is S sub n = n * (a sub 1 + a sub n) / 2, where n is the number of terms, a sub 1 is the first term, and a sub n is the last term.

What is the alternative formula for the sum of an arithmetic series when the common difference is known?

The alternative formula is S sub n = n / 2 * (2 * a sub 1 + (n - 1) * d), where n is the number of terms, a sub 1 is the first term, and d is the common difference.

How do you choose between the two arithmetic series formulas?

You use the first formula when the first and last terms of the series are given. You use the second formula when you know the first term and the common difference, but not the last term.

In the first problem, what is the value of the sum of the first 20 terms when a sub 1 is 5 and a sub 20 is 62?

The sum of the first 20 terms is 670.

How was the sum of the first 20 terms in the first problem calculated?

The sum was calculated using the formula S sub n = n * (a sub 1 + a sub n) / 2. Substituting the values, it became S sub 20 = 20 * (5 + 62) / 2, which equals 670.

In the second problem, how was the common difference calculated?

The common difference was calculated by subtracting consecutive terms in the sequence. For example, 5 - 2 = 3, 8 - 5 = 3, and so on, giving a common difference of 3.

What is the sum of the first 40 terms in the sequence 2, 5, 8, 11...?

The sum of the first 40 terms is 2,420.

How was the sum of the first 40 terms in the second problem calculated?

The sum was calculated using the formula S sub n = n / 2 * (2 * a sub 1 + (n - 1) * d). Substituting the values, S sub 40 = 40 / 2 * (2 * 2 + 39 * 3), which results in 2,420.

Outlines

00:00

🧮 Introduction to Arithmetic Series

In this introduction, Turgon begins by explaining the concept of an arithmetic series, which is defined as the partial sum of a number of terms in an arithmetic sequence. The video will focus on using two formulas to calculate the sum of such series, represented by 'S sub n.' The first formula is used when the first and last terms of the series are known, while the second formula is used when only the first term and the common difference are given. The distinction between these formulas is highlighted before proceeding to example problems.

05:02

📊 Solving for the Sum of the First 20 Terms

The first example problem involves finding the sum of the first 20 terms of an arithmetic series, where the first term is 5 and the 20th term is 62. Turgon explains that the first formula is appropriate since both the first and last terms are given. He walks through the step-by-step calculation, starting by plugging the values into the formula: Sₙ = n * (a₁ + aₙ) / 2. After simplifying, the sum of the first 20 terms is found to be 670.

🔢 Finding the Sum of the First 40 Terms Using a Different Formula

The second example problem calculates the sum of the first 40 terms of an arithmetic sequence where the terms follow a pattern (2, 5, 8, 11, ...). Since the last term is not provided, Turgon uses the second formula: Sₙ = n/2 * (2a₁ + (n-1)d), where 'd' is the common difference. He identifies the common difference as 3 and substitutes all the values. After breaking down the arithmetic, the sum of the first 40 terms is calculated to be 2,420.

📚 Conclusion and Encouragement to Subscribe

Turgon concludes the video by summarizing the two example problems and reiterating the use of both formulas for calculating the sum of arithmetic series. He encourages viewers to practice solving similar problems using the techniques covered and invites them to like and subscribe to his channel for more tutorials. Turgon ends with a friendly reminder to hit the notification bell for updates on future videos.

Keywords

💡Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence, where the difference between consecutive terms is constant. The video focuses on calculating the sum of such series using specific formulas. For instance, the script discusses finding the sum of the first 20 terms of a sequence.

💡S Sub n

S Sub n represents the sum of the first n terms in an arithmetic series. It is central to the video's explanation of how to compute the sum of a series. The formula S Sub n = n * (a1 + an) / 2 is used in the script to find the sum of terms in different sequences.

💡First Term (a1)

The first term, denoted as a1, is the initial value in an arithmetic sequence. In the video, a1 is crucial for solving the problems, such as when the first term of the sequence is given as 5. The video uses this value in various formulas to calculate sums.

💡Last Term (an)

The last term, denoted as an, is the final value in a sequence when the number of terms is finite. It is used in the formula for calculating the sum of a series. For example, in the first problem, the last term a20 is given as 62, and this is incorporated into the sum formula.

💡Number of Terms (n)

The variable n refers to the total number of terms in the arithmetic series being summed. The video highlights how to use n in formulas like S Sub n = n * (a1 + an) / 2, such as when calculating the sum of the first 20 or 40 terms.

💡Common Difference (d)

The common difference, denoted as d, is the constant difference between consecutive terms in an arithmetic sequence. In the second problem in the video, the common difference is 3, which is calculated by subtracting consecutive terms (e.g., 5 - 2 = 3). It plays a key role in formulas where the last term is not given.

💡Formula for Arithmetic Series

The video explains two key formulas for summing arithmetic series. The first is used when the first and last terms are known, while the second is used when the first term and common difference are given. The distinction between these formulas is essential for solving different types of problems.

💡Partial Sum

A partial sum refers to the sum of the first n terms in a sequence, rather than the entire series. In the video, the presenter solves problems by calculating partial sums, such as finding the sum of the first 20 or 40 terms of an arithmetic sequence.

💡Simplification

Simplification refers to reducing expressions to simpler forms for easier calculation. In the video, the presenter demonstrates this by simplifying the product and division steps in the formula, such as reducing 20/2 to 10 before multiplying it by other terms.

💡Arithmetic Sequence

An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. The video uses examples like the sequence 2, 5, 8, 11 to show how to apply the common difference and other variables in the formulas for finding the sum of terms.

Highlights

Introduction to arithmetic series as the partial sum of an arithmetic sequence.

Explanation of two formulas for finding the sum of an arithmetic series (S Sub n = n * (a₁ + aₙ) / 2 and S Sub n = n / 2 * (2a₁ + (n - 1) * d)).

Formula 1 is used when both the first and last terms of the series are known.

Formula 2 is used when the first term and the common difference are known, but the last term is not.

Example 1: Finding the sum of the first 20 terms where a₁ = 5 and aₙ = 62.

Demonstration of using Formula 1: Sₙ = n * (a₁ + aₙ) / 2 for the first 20 terms.

Calculation steps: 20 * (5 + 62) / 2, simplifying to 670.

Conclusion: The sum of the first 20 terms in the series is 670.

Example 2: Finding the sum of the first 40 terms for the series 2, 5, 8, 11...

Demonstration of using Formula 2: Sₙ = n / 2 * (2a₁ + (n - 1) * d) for the first 40 terms.

Identification of variables: a₁ = 2, n = 40, d = 3 (common difference).

Calculation steps: 40 / 2 * (2 * 2 + (40 - 1) * 3), simplifying to 2420.

Conclusion: The sum of the first 40 terms in the series is 2420.

Recap of how to choose between the two formulas depending on the given information.

Encouragement to subscribe for future arithmetic and math tutorial videos.

Transcripts

hi guys it's me turgon in our today's

video we will talk about the arithmetic

series

automatic series is considered as the

partial sum of a given number

or a given number of terms

in a given arithmetic sequence so right

now we have here two different formulas

that we will be using in this video

wherein we have here S Sub n

this S Sub n stands for

the sum

of a given arithmetic series or some of

the terms in a given analytic sequence

so for the first Formula S Sub n is

equal to n times a sub 1 plus a sub n

over 2. where inner n is the number of

terms a sub 1 is the first term a sub n

is considered as the last term in a

given series

for the next Formula

we have your a sub S Sub n is equal to n

over 2. times 2 a sub 1 plus n minus 1

times d

so the difference between these two

formula or concrete in the meeting is

that you can use the first Formula if

given the universe and last term of the

given series here in a man

uh you can use this formula if I'm given

Lang I a sub 1 and n and your common

difference so let's solve a problem

we have here find the sum of the first

20 terms so we have here first 20 terms

[Music]

first 20 terms of a given series were in

term our first term is five well a sub

20 is 62. now in this case guys

your a sub 1

is equal to five

this a sub 20 will be considered as the

last term in a given series meaning your

a sub 20

is also equal to a sub m

and that is equal to 62 meaning

if we will choose among these two

formula

much better to use S Sub n okay let's

write down the formula again

in this paper

we have S Sub n

is equal to n

times

a sub 1 plus a sub n over

two

so what's next

is that we will use this formula this s

of n will become S Sub 20. because all

we need to do is to get the sum of the

first 20 terms so this is 20. and then

year n here is equal to 20. meaning for

this variable n it will become 20

times

your a sub 1 which is 5.

plus your a sub n or a sub 20 or a sub

20 is equal to

62 over

2.

okay simplify this

it will become

20

times 5 plus 6 is 2

that is equal to

67

over 2. so as you can see guys

when you multiply this to numerator

numerator tapos your denominator is here

is one one times two it will become

20 times

six over seven a six seven over two

we're in

instead of multiplying 20 by 67 much

better if we will simplify first twenty

and two so we can cancel out two cancel

at point it will become ten so what we

have now is simply 10 plus 6 times 10

times 67 meaning

your S Sub 20 or the sum of the first 20

terms of the given sequence are series

or in the first term

in the 20th term 62. their sum is equal

to

670 because we have 10 times 67 and this

is the answer for the first problem

now let's continue

for the second problem guys here's the

second problem

what we have here is

we need to find the sum of the first 40

terms of the arithmetic Series 2 5 8 and

11 and so on now as you can see

we only have here

a sub 1 we don't have the last term of

the given sequence meaning

among the two different formulas that we

have kanina

we will be using this formula okay

so I will try to rewrite the formula we

have

S Sub n is equal to n over 2

times

2

a sub 1

plus n minus 1

times d

okay so let's list down all the needed

variables here your a sub 1 is

definitely two

now for the variable n

as you can see we have here first 40

terms meaning your n is equal to 40.

for the common difference d

so you can easily identify this one

because this one is an easy type of

arithmetic sequence

5 minus two is three eight minus five is

three eleven minus eight is three

therefore

their common difference is three now

after getting these variables we are now

ready to use this formula

your S Sub n will become S Sub 20

Over N over 2

ah sorry this is 40 guys S Sub 40 my

fault guys

this is Asap 40

is equal to 40 over 2

times

2 times 2 times your a sub 1 is 2 this

is 2 plus

we have here n minus one

here n is 40

then you have minus 1 here times d

which is equal to 3.

simplify first we have S Sub 4 d

40 divided by 2 is 20.

now let me use the parentheses 2 times 2

is 4.

Plus

40 minus 1 is 39 so we have 39

times

3.

okay so what we have now first is to

simplify this

you're 39

times 3 so we have 3 times 10 which is

27 so this is 7 then carry two

three times three is nine plus two which

is eleven meaning

this is

117 so we have S of 40.

is equal to 20

times 4 plus 117 right

and this one

we have 20 times 4 plus 17 which is

equal to one hundred twenty one

so what we need to do now is to multiply

this

so your S Sub 40

is equal to 121

times 20.

bring down zero two times one is two

two times two is four

one times one is two

meaning

the sum of the first 40 terms of the

given series is two thousand four

hundred

twenty

okay guys

so I hope guys to learn something from

this video on how to do

the sum of the given arithmetic series

using these two formulas so I hope you

like this video

so if you're new to my channel don't

forget to like And subscribe but hit the

Bell button for you to be updated latest

uploads again it's me teacher gone

bye