InterviewSolution
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InterviewSolution
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Prove that `1^1xx2^2xx3^3xx...xn^nlt ((2n+1)/(3))^((n(+1))/(2)` |
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Answer» `1^(2) = 1` `2^(2) = 2 + 2` `3^(2) = 3 + 3 + 3` `:.` Using weighted means `(1 + (2 + 2) + (3 + 3 + 3) + ….. + (n + n + ….n "times"))/(1 + 2 + 3 + …..+ n)` `ge (1^(1). 2^(2)……. N^(n))^((1)/(1 + 2 + 3 + … + n))` `implies (1 + 2^(2) + 3^(2) + .... + n^(2))/((n(n + 1))/(2)) ge (1^(1) 2^(2)...... n^(n))^((2)/(n(n + 1)))` `implies (n(n + 1)(2n + 1)/(6))/((n(n + 1))/(2)) ge (1^(1). 2^(2).... n^(n))^(n(n + 1))` `implies (2n + 1)/(3) ge (1^(1) 2^(2) .... n^(2))^((2)/(n(n + 1))` `implies 1^(1). 2^(2). 3^(3) .... n^(n) le ((2n + 1)/(3))^((n(n + 1))/(2))` |
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