Question

If the sum of the first ten terms of the series,

$open parentheses 1 3 over 5 close parentheses squared space plus open parentheses 2 2 over 5 close parentheses squared space plus open parentheses 3 1 fifth close parentheses squared space plus space 4 squared space plus space open parentheses 4 4 over 5 close parentheses squared space plus.....$ is 16/5 m, the m is equal to

102

101

100

99

Solution

101

Let S₁₀ be the sum of first ten terms of the series. Then, we have

$S subscript 10 space equals open parentheses 1 3 over 5 close parentheses squared space plus open parentheses 2 2 over 5 close parentheses squared space plus open parentheses 3 1 fifth close parentheses squared space plus space 4 squared space plus space open parentheses 4 4 over 5 close parentheses squared space plus..... space to space 10 space terms equals open parentheses 8 over 5 close parentheses squared space plus open parentheses 12 over 5 close parentheses squared space plus open parentheses 16 over 5 close parentheses squared space plus 4 squared space plus open parentheses 24 over 5 close parentheses squared space plus.... space to space 10 space terms equals 4 squared over 5 squared space left parenthesis 2 squared space plus 3 squared plus 4 squared space plus 5 squared plus........ space to space 10 space terms right parenthesis equals 4 squared over 5 squared space left parenthesis 2 squared space plus 3 squared plus 4 squared space plus 5 squared plus...11 squared right parenthesis equals space 16 over 25 space left parenthesis 1 squared space plus 2 squared space plus..... plus 11 squared right parenthesis minus 1 squared right square bracket equals 16 over 25 space open parentheses fraction numerator 11. left parenthesis 11 plus 1 right parenthesis left parenthesis 2.11 plus 1 right parenthesis over denominator 6 end fraction minus 1 close parentheses space equals space 16 over 25 left parenthesis 506 minus 1 right parenthesis space equals space 16 over 25 space straight x space 505 rightwards double arrow 16 over 5 space straight m space equals space 16 over 25 space straight x space 505 straight m space equals space 101$

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Permutations And Combinations

Some More Questions From Permutations and Combinations Chapter

Determine K, so that K + 2, 4K – 6 and 3K – 2 are three consecutive terms of an A.P.

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