Question

The first and last terms of an $AP$ are a and $l$ respectively. If $S$ be the sum of all the terms of the $AP$, then common difference is:

• A. $\frac{l^2-a^2}{2S-(l+a)}$
• B. $\frac{l^2-a^2}{2S-(l-a)}$
• C. $\frac{l^2+a^2}{2S+(l+a)}$
• D. $\frac{l^2+a^2}{2S-(l+a)}$

Answer 1

Answer: (A)

If a be the first term and $l$ be the last term of an $AP$, then $S=\frac{n}{2}(a+l)$.
Let $d$ be the common difference of an $AP$, then $S=\frac{n}{2}(a+l)...$(i)
Also $l=a+(n-1)d$
$\Rightarrow d=\frac{l-a}{n-1}$
$\Rightarrow d=\frac{l-a}{\frac{2S}{a+l}-1}[$ from (i)]
$=\frac{(l-a)(a+l)}{2S-a-l}$
$=\frac{l^2-a^2}{2S-a-l}$