First slide

Harmonic Progression

Question

Suppose $a, b, c,$ are in A.P. and $- 1 < a, b, c,$ Let $x = \sum_{n = 0}^{\infty} a^{n}, y = \sum_{n = 0}^{\infty} b^{n}, z = \sum_{n = 0}^{\infty} c^{n},$ then $x, y, z$ are in

Moderate

A

A.P.
B

G.P.
C

H.P.
D

none of these

Solution

We have $x = \frac{1}{1 - a}, y = \frac{1}{1 - b}, z = \frac{1}{1 - c} .$
As $a, b, c$ are in A.P., $- a, - b, - c$ are in A.P. $\Rightarrow 1 - a, 1 - b, 1 - c$ are in A.P.
$\Rightarrow \frac{1}{1 - a}, \frac{1}{1 - b}, \frac{1}{1 - c}$ are in H.P.

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