Find the sum of n^th term of an series whose n^th term is given by n^2 (n^2-1).

Asked by nikhilrathour52 | 29th May, 2019, 08:31: AM

Expert Answer:

S u m space o f space n space t e r m s space o f space t h e space s e r i e s space w h o s e space n t h space t e r m space i s space n squared open parentheses n squared minus 1 close parentheses space i s space g i v e n space b y
equals sum from k equals 1 to n of k squared open parentheses k squared minus 1 close parentheses
equals sum from k equals 1 to n of open parentheses k to the power of 4 minus k squared close parentheses
equals sum from k equals 1 to n of k to the power of 4 minus sum from k equals 1 to n of k squared
equals fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses open parentheses 3 n squared plus 3 n minus 1 close parentheses over denominator 30 end fraction minus fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 6 end fraction
equals fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses open parentheses 3 n squared plus 3 n minus 1 close parentheses minus 5 n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 30 end fraction
equals fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses open square brackets 3 n squared plus 3 n minus 1 minus 5 close square brackets over denominator 30 end fraction
equals fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses open square brackets n squared plus n minus 2 close square brackets over denominator 30 end fraction
equals fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses open parentheses n plus 2 close parentheses left parenthesis n minus 1 right parenthesis over denominator 30 end fraction
equals fraction numerator n open parentheses n squared minus 1 close parentheses open parentheses n plus 2 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 30 end fraction

Answered by Renu Varma | 29th May, 2019, 10:42: AM

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