Arithmetic Series - Sum of the Terms of Arithmetic Sequence
Summary
TLDRIn 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
🧮 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.
📊 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.
Mindmap
Keywords
💡Arithmetic Series
💡S Sub n
💡First Term (a1)
💡Last Term (an)
💡Number of Terms (n)
💡Common Difference (d)
💡Formula for Arithmetic Series
💡Partial Sum
💡Simplification
💡Arithmetic Sequence
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
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