If \mathop {\lim }\limits_{n o \infty } \,\,{{{1^a} + {2^a} + ...... + {n^a}} \over {{{(n + 1)}^{a - 1}}\left[ {\left( {na + 1} ight) + \left( {na + 2} ight) + ..... + \left( {na + n} ight)} ight]}} = {1 \over {60}} for some positive real number a, then a is equal to :

Question

If \mathop {\lim }\limits_{n 	o \infty } \,\,{{{1^a} + {2^a} + ...... + {n^a}} \over {{{(n + 1)}^{a - 1}}\left[ {\left( {na + 1} ight) + \left( {na + 2} ight) + ..... + \left( {na + n} ight)} ight]}} = {1 \over {60}} for some positive real number a, then a is equal to :

Accepted Answer

Answer: A. 7 \mathop {\lim }\limits_{n 	o \infty } {{{1 \over {\left( {a + 1} ight)}}\,.\,{n^{a + 1}} + {a_1}{n^a} + {a_2}{n^{a - 1}} + .\,.\,.\,.} \over {{{\left( {n + 1} ight)}^{a - 1}}.\,{n^2}\left( {a + {{1 + {1 \over n}} \over 2}} ight)}} = {1 \over {60}} \Rightarrow \mathop {\lim }\limits_{n 	o \infty } {{{{\left( {{1 \over n}} ight)}^2} + {{\left( {{2 \over n}} ight)}^a} + ....... + {{\left( {{n \over n}} ight)}^a}} \over {{{\left( {n + 1} ight)}^{a - 1}}\left[ {{a^2}a + {{n\left( {n + 1} ight)} \over 2}} ight]}} = {1 \over {60}} = {{\mathop {\lim }\limits_{n 	o \infty } {1 \over n}\sum\limits_{r = 1}^n {{{\left( {{r \over n}} ight)}^a}} } \over {{{\left( {1 + {1 \over n}} ight)}^{a - 1}}\left[ {a + {1 \over 2}\left( {1 + {1 \over n}} ight)} ight]}} = {1 \over {60}} = {{\int_0^1 {{x^a}dx} } \over {\left( {a + {1 \over 2}} ight)}} = {1 \over {60}} = {{{1 \over {a + 1}}} \over {a + {1 \over 2}}} = {1 \over {60}} \Rightarrow {{{1 \over {a + 1}}} \over {\left( {a + {1 \over 2}} ight)}} = {1 \over {60}} \Rightarrow \left( {a + 1} ight)\left( {2a + 1} ight) = 120 \Rightarrow 2a2 + 3a - 119 = 0 \Rightarrow 2a2 + 17a - 14a - 119 = 0 \Rightarrow (a - 7) (2a + 17) = 0 \Rightarrow a = 7, - {{17} \over 2}

If $$\mathop {\lim }\limits_{n \to \infty } \,\,{{{1^a} + {2^a} + ...... + {n^a}} \over {{{(n + 1)}^{a - 1}}\left[ {\left( {na + 1} \right) + \left( {na + 2} \right) + ..... + \left( {na + n} \right)} \right]}} = {1 \over {60}}$$ for some positive real number a, then a is equal to :

Answer

Explanation

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