[x = _n = 0^ a^n,y = _n = 0^ b^n,z = _n = 0^ c^n ], where a, b, c are in A.P.

Question

$$x = \sum\limits_{n = 0}^\infty  {{a^n},y = \sum\limits_{n = 0}^\infty  {{b^n},z = \sum\limits_{n = 0}^\infty  {{c^n}} } } $$, where a, b, c are in A.P. and |a|

Accepted Answer

$$x = \sum\limits_{n = 0}^\infty  {{a^n} = {1 \over {1 - a}};\,y = \sum\limits_{n = 0}^\infty  {{b^n} = {1 \over {1 - b}};\,z = \sum\limits_{n = 0}^\infty  {{c^n} = {1 \over {1 - c}}} } } $$

Now,

a, b, c $$\to$$ AP

1 - a, 1 - b, 1 - c $$\to$$ AP

$${1 \over {1 - a}},\,{1 \over {1 - b}},\,{1 \over {1 - c}} \to HP$$

x, y, z $$\to$$ HP

$$\therefore$$ $${1 \over x},{1 \over y},{1 \over z} \to AP$$

Solutions · 1