What is the Sum of an Arithmetic Sequence?

In summary, the conversation discusses the concept of arithmetic series and its relation to arithmetic sequences. The formula for finding the sum of an arithmetic series is derived and used to find the sum of the first four and first 100 terms in the given sequence. The distinction between an arithmetic series and an arithmetic sequence is clarified.
  • #1
karush
Gold Member
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I am new to this topic so...

Let $S_n$ be the sum of the first $n$ terms of the arithmetic series $2+4+6+...$

this one looks simple $S_n=2+2n$

Find $S_4$ and $S_{100}$

$S_4=2+2(4)=9$
$S_100=2+2(100)=202$

is arithmetic series and arithmetic sequence the same thing?:cool:
 
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  • #2
Re: simple arthmetic series

Series is sum of elements

Suppose that

\(\displaystyle S_n = 1+2+3+ \cdots +n =\frac{n(n+1)}{2}\) (1)

$S_n$ is the sum of the nth elements . So for example $S_4 = 1+2+3+4 = 10$

$S_n$ itself contains a sequence of elements because for each index $n$ we can find
$S$ .

Notice that your series is (1) multiplied by 2 .
 
  • #3
Re: simple arthmetic series

ZaidAlyafey said:
Series is sum of elements

Suppose that

\(\displaystyle S_n = 1+2+3+ \cdots +n =\frac{n(n+1)}{2}\) (1)

$S_n$ is the sum of the nth elements . So for example $S_4 = 1+2+3+4 = 10$

$S_n$ itself contains a sequence of elements because for each index $n$ we can find
$S$ .

Notice that your series is (1) multiplied by 2 .

I see so I found the nth term not the sum of the terms..
 
  • #4
Re: simple arthmetic series

You have got some mistakes because that doesn't even find the nth elements that should be

\(\displaystyle S_k = \sum_{n=1}^{k}a_n= 2+4+6+\cdots+k \)

Now you can write $a_n = 2n$
 
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  • #5
Re: simple arthmetic series

$\displaystyle S_4 = \sum_{n=1}^{4}(2n+2)= 28$

but I thot $2+4+6+8=20$
 
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  • #6
Re: simple arthmetic series

karush said:
$\displaystyle S_4 = \sum_{n=1}^{4}(2n+2)= 28$

but I thot $2+4+6+8=20$

Yes, that should be $a_n = 2n$
 
  • #7
Re: simple arthmetic series

karush said:
I am new to this topic so...

Let $S_n$ be the sum of the first $n$ terms of the arithmetic series $2+4+6+...$

this one looks simple $S_n=2+2n$

Find $S_4$ and $S_{100}$

$S_4=2+2(4)=9$
$S_100=2+2(100)=202$

is arithmetic series and arithmetic sequence the same thing?:cool:

Perhaps the simplest way to derive the formula for the sum of an arithmetic series is to write the series "frontwards" and "backwards" then add term by term. In this case, we could write:

\(\displaystyle S_n=2+4+6+\cdots+2(n-2)+2(n-1)+2n\)

\(\displaystyle S_n=2n+2(n-1)+2(n-2)+\cdots+6+4+3\)

Adding, we find:

\(\displaystyle 2S_n=(2n+2)+(2n+2)+(2n+2)+\cdots+(2n+2)+(2n+2)+(2n+2)\)

Notice we have $n$ identical terms on the right, so we may write:

\(\displaystyle 2S_n=n(2n+2)=2n(n+1)\)

Dividing through by 2, we obtain:

\(\displaystyle S_n=n(n+1)\)

And now we have the formula suggested by Zaid. (Sun)
 
  • #8
Re: simple arithmetic series

karush said:
I am new to this topic so...

Let $S_n$ be the sum of the first $n$ terms of the arithmetic series $2+4+6+...$

this one looks simple $S_n=2+2n$
I take it you are starting the sum at n= 0 (which does NOT actually give the first "n" terms of the sequence) so that $S_0= 2+ 2(0)= 2$, but then $S_1= 2+ 2(1)= 4$ which is the next term but NOT the sum of the first two terms of the sequence, which is 6. that is clearly NOT correct.

Find $S_4$ and $S_{100}$

$S_4=2+2(4)=9$
$S_100=2+2(100)=202$

is arithmetic series and arithmetic sequence the same thing?:cool:
An arithmetic series is a sum while a sequence is just the numbers themselves. That is "2, 4, 6, 10, 12, ..." is an arithmetic sequence while 2+ 4+ 6+ 10+ 12+ ... is an arithmetic series.

(Do you not have a textbook that defines these things?)
 
  • #9
Re: simple arithmetic series

HallsofIvy said:
I take it you are starting the sum at n= 0 (which does NOT actually give the first "n" terms of the sequence) so that $S_0= 2+ 2(0)= 2$, but then $S_1= 2+ 2(1)= 4$ which is the next term but NOT the sum of the first two terms of the sequence, which is 6. that is clearly NOT correct. An arithmetic series is a sum while a sequence is just the numbers themselves. That is "2, 4, 6, 10, 12, ..." is an arithmetic sequence while 2+ 4+ 6+ 10+ 12+ ... is an arithmetic series.

(Do you not have a textbook that defines these things?)

yes, i am using Sullivans, Algrebra & Trigonometry, but they call "arithmetic series" "Sum of a Sequence" which appears to be the same idea$\displaystyle S_4 = \sum_{n=0}^{4}(2n)= 20$

$\displaystyle S_{100} = \sum_{n=0}^{100}(2n)= 10100$
 
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Related to What is the Sum of an Arithmetic Sequence?

What is a simple arithmetic series?

A simple arithmetic series is a sequence of numbers where each term is obtained by adding a constant number to the previous term. For example, the series 2, 5, 8, 11, ... is a simple arithmetic series with a common difference of 3.

What is the formula for finding the sum of a simple arithmetic series?

The formula for finding the sum of a simple arithmetic series is Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, and d is the common difference.

How do you find the common difference in a simple arithmetic series?

The common difference in a simple arithmetic series can be found by subtracting any term from the term that follows it. For example, in the series 3, 7, 11, 15, ... the common difference is 4 (7-3 = 4, 11-7 = 4, etc.).

What is the nth term in a simple arithmetic series?

The nth term in a simple arithmetic series can be found using the formula tn = a + (n-1)d, where tn is the nth term, a is the first term, and d is the common difference. For example, in the series 2, 5, 8, 11, ... the 6th term would be 17 (2 + (6-1)3 = 17).

How can simple arithmetic series be used in real life?

Simple arithmetic series can be used in real life for a variety of tasks, such as calculating interest rates, predicting population growth, and determining the total cost of a purchase over time. They can also be used in fields such as economics, physics, and engineering to model and analyze patterns and trends.

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