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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.

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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.

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.

Let the four numbers in G.P. be $space space straight a over straight r cubed space straight a over straight r comma space space ar comma space space ar cubed$

Since their product is 4096.

∴ $open parentheses straight a over straight r cubed close parentheses space open parentheses straight a over straight r close parentheses space left parenthesis ar right parenthesis space left parenthesis ar cubed right parenthesis space equals space 4096 space rightwards double arrow space space space straight a to the power of 4 space equals space 4096 space space rightwards double arrow space straight a to the power of 4 space equals space 8 to the power of 4 space rightwards double arrow space straight a space equals space 8$

Since their sum is 85

∴ $straight a over straight r cubed plus straight a over straight r plus ar plus ar cubed space equals space 85 space rightwards double arrow space space straight a open parentheses straight r cubed plus 1 over straight r cubed close parentheses plus straight a open parentheses straight r plus 1 over straight r close parentheses space equals space 85$

$rightwards double arrow$ $8 open square brackets open parentheses straight r plus 1 third close parentheses cubed minus space 3 straight r 1 over straight r open parentheses straight r plus 1 over straight r close parentheses close square brackets plus 8 open parentheses straight r plus 1 over straight r close parentheses space equals space 85$

Put $straight r plus 1 over straight r space equals straight t$
∴ $8 straight t cubed minus 24 straight t plus 8 straight t space equals space 85 space space space space space space or space space space space space 8 straight t cubed minus 16 straight t minus 85 space equals space 0$ or

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Putting 2t = x, we get

 $straight x cubed minus 8 straight x minus 85 equals 0$

By inspection x = 5 satisfies the equation. So x - 5 is a factor of $straight x cubed minus 8 straight x minus 85$ . Other roots are given by

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$rightwards double arrow$ $straight x equals space fraction numerator negative 5 plus-or-minus square root of 25 minus 68 end root over denominator 2 end fraction$

$rightwards double arrow$ x is imaginary

∴ x = 5 is the only real value

So, 2t = 5,

$space space space rightwards double arrow space space space space straight t space equals space 5 over 2 comma space space straight r plus space 1 over straight r space equals space 5 over 2$

$space space space rightwards double arrow space space space space$ $2 straight r squared minus 5 straight r plus 2 space equals space 0$

$straight r equals space fraction numerator 5 plus-or-minus square root of 25 minus 16 end root over denominator 4 end fraction space equals space fraction numerator 5 plus-or-minus 3 over denominator 4 end fraction space equals space 2 comma space space 1 half$
When a = 8, r = 2, numbers are $8 over 2 cubed comma space 8 over 2 comma space 8 left parenthesis 2 right parenthesis. space 8. space 2 cubed space space or space space space 1 comma space 4 comma space 16 comma space 64$

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