A.P problem gaussian sum (average of terms explanation)

Question

The question is "the sum of first n terms of an arithmetic progression,if the last term is given $S_n=\frac{n(a+l)}{2}$; what does "$\frac{a+l}{2}$" represent?

Answer 1

let $S=a+(a+k)+(a+2k)...(a+nk)$ be the sum you want. clearly $S=(a+nk)+(a+(n-1)k)+...a+k+a$ ( the same but in reverse order) add top to bottom to get:

$2S=(2a+nk)+(2a+nk)....(2a+nk)=n(2a+nk)\rightarrow S=\frac{n(2a+nk)}{2}$

n is the number of terms, so that means $\frac{2a+nk}{2}$ is the arithmetic mean of the terms and also the sum of the first and last term

Answer 2

They are using $a$ as the first term and $l$ as the last term, then asking you to prove that the sum of the first $n$ terms is $n\frac {a+l}2$, the product of the number of terms and the average term.

Answer 3

S = sum

n = number of terms

a = first term

l = last term

(a+l)/2 = average of first and last term.

This makes sense for arithmetic sequences. Example:

{1, 3, 5, 7, 9}

n = 5; a = 1; l = 9;

Sum: by adding: 1+3+5+7+9 = 25. By equation: (5)(1+9)/2 = 25.