MHB What is the Sum of an Arithmetic Sequence?

[karush]

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?

 

[alyafey22]

Re: simple arthmetic series

Series is sum of elements

Suppose that

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

 

[karush]

Re: simple arthmetic series

Series is sum of elements

Suppose that

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

 

[alyafey22]

Re: simple arthmetic series

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

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

Now you can write $a_n = 2n$

 

[karush]

Re: simple arthmetic series

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

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

 

[alyafey22]

Re: simple arthmetic series

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

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

Yes, that should be $a_n = 2n$

 

[MarkFL]

Re: simple arthmetic series

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?

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:

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

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

Adding, we find:

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

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

$$2S_n=n(2n+2)=2n(n+1)$$

Dividing through by 2, we obtain:

$$S_n=n(n+1)$$

And now we have the formula suggested by Zaid. (Sun)

 

[HallsofIvy]

Re: simple arithmetic series

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?

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?)

 

[karush]

Re: simple arithmetic series

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$\displaystyle S_4 = \sum_{n=0}^{4}(2n)= 20$

$\displaystyle S_{100} = \sum_{n=0}^{100}(2n)= 10100$