A further explanation of the origin of this problem

I see your thinking there, but it doesn't work. I'll explain the full problem, then.

I was given a formula that could be used to find the nth term in an arithmetic sequence. This equation was:

Where:

=answer

n=the term you are looking for (e.g. the 5th term, 12th term, or 78th term)

b=the number of terms shown

d=1st difference

a=the last term of the sequence shown

I was asked to prove this is true in all cases, and to do so, I took the already proven theorem of:

Where:

=answer

= first term of the sequence

d=1st difference

n=term you are looking for

and set about turning

into

The first problem I ran into was the fact that there was no in the first equation. I determined (but have yet to prove) that dividing the last number shown (a) by the number of terms shown (b), you always get the first term. Thus, my target equation can be written as

I then set about trying to reach that equation from my first, and this is where I got stuck. I may have gone about this wrong, and, if so, please correct me. I hope the extra information helped.(Happy)