Let, $X = 1 + 3 a + 6 a^{2} + 10 a^{3} + . . . ., | a | < 1$
$y = 1 + 4 b + 10 b^{2} + 20 b^{3} + . . . ., | b | < 1$
Find S = 1 + 3 (ab) + 5 (ab)^2 + .......
in terms of x and y

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Solution

x = 1 + 3a + $6 a^{2}$ + $10 a^{3}$ + ...
$∴$ ax = a + $3 a^{2}$ + $6 a^{3}$ + ...
$∴$ x(1 - a) = 1 + 2a + $3 a^{2}$ + $4 a^{3}$ + ...
above is A.G.S. whose $S_{\infty} = \frac{A}{1 - R} + \frac{d R}{(1 - R)^{2}}$
$∴ x (1 - a) = \frac{1}{(1 - a)^{3}} + \frac{a}{(1 - a)^{2}} + \frac{1}{(1 - a)^{2}}$
$∴ x = \frac{1}{(1 - a)^{3}}$
$∴ (a - a)^{3} = x^{- 1}$
or a = 1 - $x^{- 1 / 4}$
$∴ S = \frac{1}{1 - a b} + \frac{2 a b}{(1 - a b)^{2}},$ as above for A.G.S.
$= \frac{1 + a b}{(1 - a b)^{2}} .$
Now put the value of a and b from above in terms of x and y.

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