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. 1    
  2. fraction numerator 1 over denominator 2 end fraction    
  3. fraction numerator 2 over denominator 3 end fraction    
  4. fraction numerator 3 over denominator 2 end fraction    

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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