Find a formula in $n, a, m, d$ for the sum: $(a+md)+(a+(m+1)d+(a+(m+2)d)+ ...+ (a+(m+n)d$

Question

where m and n are integers, $n$ greater than or equal to $0$ and $a$ and $d$ are real numbers. Justify your result.

By distributing and factoring, I have gotten up to $(a+md)n+d(1+...+n)$ but I am stuck on what to do next.

Answer 1

Hint:

Change your variables!

Set $A:= a+md$ , $D:=d$ and $N:=n$. Your sequence turns out to be

$A,A+D,A+2D,...,A+ND$.

Now what is the sum of an arithmetic progression?