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Maths-

General

Easy

Question

If the sum of an infinite GP and the sum of square of its term is 3, then the common ratio of the first series is

1
$\frac{1}{2}$
$\frac{2}{3}$
$\frac{3}{2}$

The correct answer is: $fraction numerator 3 over denominator 2 end fraction$

Let the GP series be $a comma a r comma blank a r to the power of 2 end exponent comma horizontal ellipsis$
According to the given condition
$a plus a r plus a r to the power of 2 end exponent plus... equals a to the power of 2 end exponent plus a to the power of 2 end exponent r to the power of 2 end exponent plus a to the power of 2 end exponent r to the power of 4 end exponent plus... equals 3$
$rightwards double arrow blank fraction numerator a over denominator 1 minus r end fraction equals fraction numerator a to the power of 2 end exponent over denominator 1 minus r to the power of 2 end exponent end fraction equals 3$
$rightwards double arrow blank a equals 3 left parenthesis 1 minus r right parenthesis$ …(i)
and $a to the power of 2 end exponent equals 3 left parenthesis 1 minus r to the power of 2 end exponent right parenthesis$ …(ii)
Eliminating $a$ from Eqs. (i) and (ii), we get
$open square brackets 3 open parentheses 1 minus r close parentheses close square brackets to the power of 2 end exponent equals 3 left parenthesis 1 minus r to the power of 2 end exponent right parenthesis$
$rightwards double arrow blank 3 open parentheses 1 minus r close parentheses equals open parentheses 1 plus r close parentheses blank left parenthesis because r not equal to 1 right parenthesis$
$rightwards double arrow 4 r equals 2 blank rightwards double arrow r equals fraction numerator 1 over denominator 2 end fraction$
$therefore blank a equals 3 open parentheses 1 minus r close parentheses equals 3 open parentheses 1 minus fraction numerator 1 over denominator 2 end fraction close parentheses equals fraction numerator 3 over denominator 2 end fraction$

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