Find the sum of first n natural numbers  OrEvaluate  from Mat

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 Multiple Choice QuestionsShort 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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 Multiple Choice QuestionsLong 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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 Multiple Choice QuestionsShort 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.
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 Multiple Choice QuestionsLong 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.
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 Multiple Choice QuestionsShort 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
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 Multiple Choice QuestionsLong Answer Type

119.

Insert 5 G.M.'s between 32 over 9 space and space 81 over 2

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 Multiple Choice QuestionsShort Answer Type

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120.

Find the sum of first n natural numbers  Or

Evaluate sum straight n space or space sum from straight k equals 1 to straight n of space straight k


sum from straight k equals 1 to straight n of space straight k space space or space space sum straight n space equals space 1 space plus space 2 space plus 3 space plus....... space plus straight n

It is an A.P. series

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∴                sum from straight k equals 1 to straight n of space straight k space or space sum straight n space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction
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