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Sequences and Series

Short Answer Type

111. If a, b, c, d are in GP., show that (a² + b² + c²) (b² + c² + d²) = (ab + bc + cd)²

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Long Answer Type

112.

If a, b, c are in G.P. and x, y are the arithmetic means of a, b and b, c respectively,

then prove that $straight a over straight x plus straight c over straight y equals 2$ and $space space 1 over straight x plus 1 over straight y equals 2 over straight b$

171 Views

Short Answer Type

113.

If a, b, c are in A.P. and x,y,z are in G.P., show that

$straight x to the power of straight b minus straight c end exponent. space straight y to the power of straight c minus straight a end exponent. space straight z to the power of straight a minus straight b end exponent space equals space 1$

138 Views

114. Find a GP. for which sum of first two terms is –4 and fifth term is 4 times the 3rd term.

146 Views

115. If A.M. and G.M. of roots of a quadratic equation are 8 and 5 respectively, then obtain the quadratic equation.

153 Views

Long Answer Type

116. If a and b are roots of x² – 3x+p = 0 and c, d are roots of x² – 12x + q = 0 where a, b, c, d form a GP., prove that (q + p) : (q – p) = 17 : 15

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117. Find four numbers in GP. whose sum is 85 and product is 4096.

477 Views

Short Answer Type

118. Find the sum of n terms of the series whose nth term is given by

$straight n squared plus 2 to the power of straight n$

Here, $straight T subscript straight n space equals space straight n squared plus 2 to the power of straight n$

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$space space space equals left parenthesis 1 squared plus 2 squared plus 3 squared plus.... plus straight n squared right parenthesis plus left parenthesis 2 to the power of 1 plus 2 squared plus.....2 to the power of straight n right parenthesis space equals space sum straight n squared plus fraction numerator 2 left parenthesis 2 to the power of straight n minus 1 right parenthesis over denominator 2 minus 1 end fraction$

{The series is 2nd bracket is G.P. with common ratio 2>1}

$equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction space plus 2 to the power of straight n plus 1 end exponent minus 2$