Thread: 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?

Let be arithmetic with common difference d.

Then





So, in general

.

You can prove this formally by mathematical induction.

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

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.