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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 $space space 1 over straight x plus 1 over straight y equals 2 over straight b$

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Short Answer Type

113.

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

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

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

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

Let a, b be roots of required quadratic equation

Given $fraction numerator straight a plus straight b over denominator 2 end fraction space equals space 8$ and $square root of ab space equals 5$

$rightwards double arrow$ a + b = 16 and ab = 25

S = sum of roots = a + b = 16.

and P = Product of roots = ab = 25

∴ Required quadratic equation is $space space space space space space straight x squared minus Sx plus straight P space equals 0$ or $straight x squared minus 16 straight x plus 25 space equals space 0.$