# Let $\;a_{r}=\int\limits_{0}^{\frac{\pi}{4}}\;tan^{r}\;x\;dx$ , then $\;a_{1}+a_{3}\;,a_{2}+a_{4}\;,a_{3}+a_{5}\;$ are in

$(a)\;AP\qquad(b)\;GP\qquad(c)\;HP\qquad(d)\;AGP$

Answer : (c) HP
Explanation : $\;a_{r}+a_{r+2}=\int\limits_{0}^{\frac{\pi}{4}}\;tan^{r}\;x\;(1+tan^{2}\;x)\;dx$
$=\int\limits_{0}^{\frac{\pi}{4}}\;tan^{r}\;x\;sec^{2}\;x\;dx$
$=\;[\frac{tan^{r+1\;x}}{r+1}]_{0}^{\frac{\pi}{4}}=\frac{1}{r+1}$
$a_{1}+a_{3}=\frac{1}{2}$
$a_{2}+a_{4}=\frac{1}{3}\quad\;a_{3}+a_{5}=\frac{1}{4}$
They are in HP.
answered Jan 21, 2014 by