求和1/1×4+1/4×7+.+1/(3n-2)×(3n+1)

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求和1/1×4+1/4×7+.+1/(3n-2)×(3n+1)

求和1/1×4+1/4×7+.+1/(3n-2)×(3n+1)
求和1/1×4+1/4×7+.+1/(3n-2)×(3n+1)

求和1/1×4+1/4×7+.+1/(3n-2)×(3n+1)
原式=(1/3)*(1/1 - 1/4) + (1/3)*(1/4 - 1/7) + (1/3)*(1/7 - 1/10)+...+
(1/3)*[1/(3n-2) - 1/(3n+1)]
=(1/3)* {[(1 - 1/4) + (1/4 - 1/7) + (1/7 - 1/10) + ...
+ [1/(3n-2) - 1/(3n+1)] }
=(1/3)* {1 + (-1/4 + 1/4) + (- 1/7 + 1/7) + (-1/10 + 1/10) + ...
+ [-1/(3n-2) + 1/(3n-2)] - 1/(3n+1)}
=(1/3)* [1 - 1/(3n+1)]
=(1/3)* 3n/(3n+1)
=n/(3n+1)

这是比较基本的裂项求和应用,希望楼主参照楼上的做法得出其中规律