Let be arithmetic with common difference d.
Then
So, in general
.
You can prove this formally by mathematical induction.
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?