Olympiad workbookIOQMSequences and Series

If _ k=1 ^N 2 k+1 (k^2+k )^2 =0.9999 then determine the value of N .

Correct answer

Step-by-step solution

array r _ k=1 ^N 2 k+1 (k^2+k )^2 = _ k=1 ^N 2 k+1 k^2(k+1)^2 = _ k=1 ^N (k+1)^2-k^2 k^2(k+1)^2 = _ k=1 ^N ( 1 k^2 - 1 (k+1)^2 )= ( 1 1^2 - 1 2^2 )+ ( 1 2^2 - 1 3^2 ) + + ( 1 N^2 - 1 (N+1)^2 ) array aligned & =1- 1 (N+1)^2 = 9999 10000 & 1 (N+1)^2 = 1 10000 = 1 (100)^2 & N+1=100 & N=99 aligned

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