What is the formula for finding the sum of an arithmatic series?

[CathyLou]

I'd really appreciate it if someone could please help me with this question as I'm really stuck on it.

Prove that the sum of the first n terms of an arithmatic series with first term a and common difference d is given by

1/2[2a + (n-1) d]

Thank you.

Cathy

 

[radou]

Well, a start would be to write the expression for the general term of an arithmetic sequence: $$a_{i} = a_{1} + (i-1)d$$, where a1 is the first term, and d is the difference.

 

[Sane]

According to Wikipedia:

Express the arithmetic series in two different ways:

$$S_n=a_1+(a_1+d)+(a_1+2d)+ \dots +(a_1+(n-2)d)+(a_1+(n-1)d)$$

$$S_n=(a_n-(n-1)d)+(a_n-(n-2)d)+ \dots +(a_n-2d)+(a_n-d)+a_n$$

Add both sides of the two equations. All terms involving ''d'' cancel, and so we're left with:

$$2S_n=n(a_1+a_n)$$

Rearranging and remembering that $$a_n = a_1 + (n-1)d$$, we get:

$$S_n=\frac{n( a_1 + a_n)}{2}=\frac{n[ 2a_1 + (n-1)d]}{2}$$

 

[CathyLou]

Thanks so much to both of you for your help.

Cathy

 

[CathyLou]

I also stuck on the rest of the question. Could someone please help? I'd be really grateful for any assistance.

A novelist begins writing a new book. She plans to write 16 pages during the first week, 18 during the second and so on, with the number of pages increasing by 2 each week.

Find, according to her plan,

(b) how many pages she will write in the fifth week

(c) the total number of pages she will write in the first five weeks

(d) Using algebra, find how long it will take her to write the book if it has 250 pages.

Thank you.

Cathy

 

[radou]

Well, if you understand what an arithmetic sequence is, then you should be able to solve the problem. Show us some work (if you get stuck), and we'll be glad to help.

 

[CathyLou]

Well, if you understand what an arithmetic sequence is, then you should be able to solve the problem. Show us some work (if you get stuck), and we'll be glad to help.

Okay, an arithmatic series is a sequence where each term is found by adding a fixed number to the previous one.

For part b I got an answer of 24 pages and for part c I got an answer of 100 pages, but I still don't know how to do part d as I just got an answer of 235 weeks and that is obviously wrong.

Cathy

 

[physicsgal]

so you have 16, 18, 20. 22, 24, 26

so you can see each number increases by 2. and the first number starts as 16.

tn = 16 + (2n) - 2.

so:
250 = 16 + (2n) -2
234 = (2n) - 2
236 = 2n
n = 118 weeks

im pretty sure that's correct. hopefully someone will double check it. I am not good in math myself.

~Amy

 

[CathyLou]

so you have 16, 18, 20. 22, 24, 26

so you can see each number increases by 2. and the first number starts as 16.

tn = 16 + (2n) - 2.

so:
250 = 16 + (2n) -2
234 = (2n) - 2
236 = 2n
n = 118 weeks

im pretty sure that's correct. hopefully someone will double check it. I am not good in math myself.

~Amy

Thanks for your help!

Cathy

 

[Max Eilerson]

so you have 16, 18, 20. 22, 24, 26

so you can see each number increases by 2. and the first number starts as 16.

tn = 16 + (2n) - 2.

so:
250 = 16 + (2n) -2
234 = (2n) - 2
236 = 2n
n = 118 weeks

im pretty sure that's correct. hopefully someone will double check it. I am not good in math myself.

~Amy

That isn't correct which should be obvious. After 5 week she will write 16 + 2(5) = 26 pages, so it can't take more than 10 weeks from then on.

 

[radou]

You're mixing the sum of the series with the members up. You have to solve $$250 = \sum_{i=1}^n a_{i} = \sum_{i=1}^n (16+(i-1)\cdot2)$$, where n is the number of weeks it takes to write 250 pages.