# Thread: arithmetic sequences: why finding the nth term works

1. ## arithmetic sequences: why finding the nth term works

I understand HOW to create a formula to find the Nth term in an arithmetic sequence. My question is WHY does it work.

I understand what the explicit formula represents and how to use it. I am just lost on why all of it works. Why can I multiply the common difference by the nth term, and then add it to the the the first term number minus the common difference and get this magical formula?

2. Let $\displaystyle a_1,a_2,a_3,...$ be arithmetic with common difference d.

Then

$\displaystyle a_2=a_1+d$
$\displaystyle a_3=a_2+d=a_1+d+d=a_1+2d$
$\displaystyle a_4=a_3+d=a_1+2d+d=a_1+3d$

So, in general

$\displaystyle a_n=a_1+(n-1)d$.

You can prove this formally by mathematical induction.

3. Short 'n sweet (d = common diff, a = 1st term):

0,3,6,9,12,15... : nth term = d(n-1)

2,5,8,11,14,17... : comparing to above, each term is higher by a

Sooooo: nth term = d(n-1) + a ; 3(6-1) + 2 = 17

4. Thank you drsteve
That makes much more sense to me now. I had one of my 7th graders ask me why today and I told them I'd figure it out tonight. Really appreciate it.