JEE MAIN - Mathematics (2005 - No. 44)
If $$x = \sum\limits_{n = 0}^\infty {{a^n},\,\,y = \sum\limits_{n = 0}^\infty {{b^n},\,\,z = \sum\limits_{n = 0}^\infty {{c^n},} } } \,\,$$ where a, b, c are in A.P and $$\,\left| a \right| < 1,\,\left| b \right| < 1,\,\left| c \right| < 1$$ then x, y, z are in
G.P.
A.P.
Arithmetic-Geometric Progression
H.P.
Explanation
$$x = \sum\limits_{n = 0}^\infty {{a^n}} = {1 \over {1 - a}}\,\,\,\,\,\,\,\,\,\,a = 1 - {1 \over x}$$
$$y = \sum\limits_{n = 0}^\infty {{b^n}} = {1 \over {1 - b}}\,\,\,\,\,\,\,\,\,\,b = 1 - {1 \over y}$$
$$z = \sum\limits_{n = 0}^\infty {{c^n}} = {1 \over {1 - c}}\,\,\,\,\,\,\,\,\,\,c = 1 - {1 \over z}$$
$$a,b,c$$ are in $$A.P.$$ OR $$2b = a + c$$
$$2\left( {1 - {1 \over y}} \right) = 1 - {1 \over x} + 1 - {1 \over y}$$
$${2 \over y} = {1 \over x} + {1 \over z} \Rightarrow x,y,z$$ are in $$H.P.$$
$$y = \sum\limits_{n = 0}^\infty {{b^n}} = {1 \over {1 - b}}\,\,\,\,\,\,\,\,\,\,b = 1 - {1 \over y}$$
$$z = \sum\limits_{n = 0}^\infty {{c^n}} = {1 \over {1 - c}}\,\,\,\,\,\,\,\,\,\,c = 1 - {1 \over z}$$
$$a,b,c$$ are in $$A.P.$$ OR $$2b = a + c$$
$$2\left( {1 - {1 \over y}} \right) = 1 - {1 \over x} + 1 - {1 \over y}$$
$${2 \over y} = {1 \over x} + {1 \over z} \Rightarrow x,y,z$$ are in $$H.P.$$
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