Question

If $a$ and $\ell$ be the first and last term of an A.P.and S be the sum of its all terms, then its common difference is.

• A. $\frac{{\ell }^2+a^2}{2S-\ell -a}$
• B. $\frac{{\ell }^2-a^2}{2S-\ell -a}$
• C. $\frac{{\ell }^2-a^2}{2S+\ell +a}$
• D. none of these

Answer 1

The correct option is B $\frac{{\ell }^2-a^2}{2S-\ell -a}$

Let the terms of A.P. are
$a,a+d,a+2d,...a+(n-1)d$
Given $S=\frac{n}{2}(a+l)\therefore n=\frac{2S}{a+l}$
Again $l=a+(n-1)d$
$\therefore l=a+(\frac{2S}{a+l}-1)d$
$\therefore d=\frac{(l-a)(l+a)}{2S-1-a}$
$d=\frac{l^2-a^2}{2S-l-a}$
Hence, option 'B' is correct.