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 $l=a+(n-1) d$
$\Rightarrow d=\frac{l-a}{n-1}$
$\Rightarrow d=\frac{l-a}{\frac{2 S}{a+l}-1}[$ from (i)]
$=\frac{(l-a)(a+l)}{2 S-a-l}$
$=\frac{l^{2}-a^{2}}{2 S-a-l}$