First slide

Harmonic Progression

Question

$Let I_{n} = \int_{0}^{π / 4} \tan^{n} xdx, then I_{2} + I_{4}, I_{3} + I_{5}, I_{4} + I_{6}, I_{5} + I_{7}, \dots are in$

Moderate

A

HP
B

GP
C

AP
D

AGP

Solution

$I_{n} + I_{n-2} = \frac{1}{n - 1}$
$I_{n+2} + I_{n} = \frac{1}{n+ 1}$
$\begin{aligned} I_{2} + I_{4} = \frac{1}{3} \\ I_{3} + I_{5} = \frac{1}{4} \\ I_{4} + I_{6} = \frac{1}{5} \end{aligned}$
$\frac{1}{3}, \frac{1}{4}, \frac{1}{5} . . . a r e H . P$

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