Let S_n= _ k=1 ^n(-1)^ k-1 k^2 for n 1 . Given that S_ 2 n =-n(2 n+1) for n=1,2,3, . then S₇₇=

Step-by-step solution

aligned & S_n= _ k=1 ^n(-1)^ k-1 k^2 & S_n=1^2-2^2+3^2-4^2+ (-1)^ n-1 n^2 & S₇₇=1^2-2^2+3^2-4^2+ (-1)⁷⁶ 77^2 & S₇₇= (1^2+2^2+3^2+ +77^2 ) & -2 (2^2+4^2+6^2+ +76^2 ) & = (1^2+2^2+3^2+ +77^2 ) & -2 2^2 (1^2+2^2+3^2+ +38^2 ) aligned aligned & = _ r=1 ⁷⁷ r^2-2 2^2 _ k=1 ³⁸ k^2 & = 77 78 155 6 - 8 38 39 77 6 & = [ as n^2= n(n+1)(2 n+1) 6 ] & =155155-152152=3003 . aligned