# Let $a_{n}\;$ be $\;n^{th}$ term of AP. If $\;\sum_{r=1}^{50}\;a_{2r}=p\;and \;\sum_{r=1}^{50}\;a_{2r-1}=q\;$ the common difference is :

$(a)\;p-q\qquad(b)\;q-p\qquad(c)\;\frac{p-q}{100}\qquad(d)\;\frac{p-q}{50}$

Answer : (d) $\frac{p-q}{50}$
Let d be common difference
$p-q=\sum_{r=1}^{50}\;(a_{2r}-a_{2r-1})=\sum_{r=1}^{50}\;d$
$=50d$
$d=\frac{p-q}{50}.$