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Maths-

General

Easy

Question

If the sum to infinity of the series $1 plus 2 r plus 3 r to the power of 2 end exponent plus 4 r to the power of 3 end exponent plus midline horizontal ellipsis$ is 9/4, then value of $r$ is

1/2
1/3
1/4
None of these

The correct answer is: 1/3

$S equals 1 plus 2 r plus 3 r to the power of 2 end exponent plus 4 r to the power of 3 end exponent plus horizontal ellipsis$
$r S equals r plus 2 r to the power of 2 end exponent plus 3 r to the power of 3 end exponent plus 4 r to the power of 4 end exponent plus horizontal ellipsis$
$rightwards double arrow open parentheses 1 minus r close parentheses S equals 1 plus r plus r to the power of 2 end exponent plus r to the power of 3 end exponent plus horizontal ellipsis$
$equals fraction numerator 1 over denominator 1 minus r end fraction$
$rightwards double arrow S equals fraction numerator 1 over denominator open parentheses 1 minus r close parentheses to the power of 2 end exponent end fraction$
Given, $S equals 9 divided by 4 blank rightwards double arrow fraction numerator 1 over denominator open parentheses 1 minus r close parentheses to the power of 2 end exponent end fraction equals 9 divided by 4$
$rightwards double arrow 1 minus r equals plus-or-minus fraction numerator 2 over denominator 3 end fraction$
$rightwards double arrow r equals 1 divided by 3$ or $5 divided by 3$
Hence, $r equals 1 divided by 3$ as $0 less than open vertical bar r close vertical bar less than 1$

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