First slide

Harmonic Progression

Question

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

Moderate

A

A.P.
B

G.P.
C

H.P.
D

none of these

Solution

We have for $r \geq 2,$
$\begin{array}{l} I_{r} + I_{r + 2} = \int_{0}^{π / 4} \tan^{r} x (1 + \tan^{2} x) d x \\ = \int_{0}^{π / 4} \tan^{r} x \sec^{2} x d x \\ {= \frac{1}{r + 1} \tan^{r + 1} x]}_{0}^{π / 4} = \frac{1}{r + 1} \end{array}$
Thus, the given sequence becomes,
$\frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \dots$
This is clearly an H.P.

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