Question

If 1st term of an A. P. is a, 2nd term is b and last term is c, then show that sum of all terms is (a+c)(b+c-2a)/2(b-a)

Answer 1

first term =a
common difference d = b-a
let total number of terms be n
then
c=a+(n-1)(b-a)
c= a+ n(b-a)-b+a
n(b-a)= c+b-2a
$n = \frac{c + b - 2a}{b - a}$
sum of all teens
$\\ = \frac{n}{2} (first \: term + last \: term) \\ = \frac{(c + b - 2a)}{2( b- a)} (a + c)$
hence provided