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  4. If the sum to infinity of the series, 1 + 4x + 7x2 + 10x3 +........, is 35/16, where | x | < 1, then x equals to

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Q. If the sum to infinity of the series, $1 + 4x + 7x^2 + 10x^3 +........,$ is $35/16$, where $| x | < 1$, then $x$ equals to

Sequences and Series

Solution:

$S = 1 + 4x + 7x^2 + 10x^3 +........$
$x.S = x + 4x^2 + 7x^3 +........$
Subtract
$S (1 - x) = 1 + 3x + 3x^2 + 3x^3 +.......$
$S\left(1-x\right)=1+3x\left(\frac{1}{1-x}\right), \because \left|x\right| < 1$
$S=\frac{1+2x}{\left(1-x\right)^{2}}$
Given : $\frac{1+2x}{\left(1-x\right)^{2}}=\frac{35}{16}$
$\Rightarrow 16 + 32x = 35 + 35x^{2} - 70x$
$\Rightarrow 35x^{2} - 102x + 19 = 0$
$\Rightarrow 35x^{2} - 7x - 95x + 19 = 0$
$\Rightarrow 7x \left(5x - 1\right) - 19 \left(5x - 1\right) = 0$
$\Rightarrow \left(5x - 1\right) \left(7x - 19\right) = 0$
$\Rightarrow x=\frac{1}{5}, \frac{19}{7}$
But $| x | < 1, \therefore x=1/5$