JEE MAIN - Mathematics (2022 - 27th June Morning Shift - No. 3)
$$x = \sum\limits_{n = 0}^\infty {{a^n},y = \sum\limits_{n = 0}^\infty {{b^n},z = \sum\limits_{n = 0}^\infty {{c^n}} } } $$, where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1, abc $$\ne$$ 0, then :
x, y, z are in A.P.
x, y, z are in G.P.
$${1 \over x}$$, $${1 \over y}$$, $${1 \over z}$$ are in A.P.
$${1 \over x}$$ + $${1 \over y}$$ + $${1 \over z}$$ = 1 $$-$$ (a + b + c)
Explanation
$$x = \sum\limits_{n = 0}^\infty {{a^n} = {1 \over {1 - a}};\,y = \sum\limits_{n = 0}^\infty {{b^n} = {1 \over {1 - b}};\,z = \sum\limits_{n = 0}^\infty {{c^n} = {1 \over {1 - c}}} } } $$
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
$$\therefore$$ $${1 \over x},{1 \over y},{1 \over z} \to AP$$
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