f(n)=cos(n*pi/2) 求f(25)+f(26)+f(27)+......+f(42)的值

f(n) + f(n+1) + f(n+2)+f(n+3)
= [f(n) + f(n+2)] + [f(n+1) + f(n+3)]
= [cosnπ/2 + cos(n+2)π/2] + [cos(n+1)π/2 + cos(n+3)π/2]
= cosnπ/2 + cos(nπ/2 + π) + cos(n+1)π/2 + cos[(n+1)π/2 + π]
= cosnπ/2 - cosnπ/2 + cos(n+1)π/2 - cos(n+1)π/2
= 0

即任意连续4项的和为0

所以
f(25)+f(26)+f(27)+···+f(42)
= f(41) + f(42)
= cos41π/2 + cos42π/2
= cosπ/2 + cosπ
= 0 - 1
= -1

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