Let-a-b-c-be-in-AP-and-a-lt-1-b-lt-1-c-lt-1-If-x-1-a-a-2-to-y-1-b-b-2-to-z-1-c-c-2-to-then-x-y-z-are-in-

Question Number 56419 by gunawan last updated on 16/Mar/19

Let a, b, c be in AP and ∣ a ∣< 1, ∣ b ∣< 1, ∣ c ∣< 1 . If

x = 1 + a + a^{2} + \dots to \infty

y = 1 + b + b^{2} + \dots to \infty

z = 1 + c + c^{2} + \dots to \infty

then x, y, z are in

Answered by tanmay.chaudhury50@gmail.com last updated on 16/Mar/19

x = \frac{1}{1 - a} y = \frac{1}{1 - b} z = \frac{1}{1 - c}

1 - a = \frac{1}{x} \to a = 1 - \frac{1}{x}

b = 1 - \frac{1}{y}

c = 1 - \frac{1}{z}

2 b = a + c

2 (1 - \frac{1}{y}) = 1 - \frac{1}{x} + 1 - \frac{1}{z}

\frac{2}{y} = \frac{1}{x} + \frac{1}{z} s o x, y a n d z i n H . P