SOLUTION: Find the sum of the first n terms of the following series 1x2x3+2x3x4+3x4x5x+---

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Question 984370: Find the sum of the first n terms of the following series
1x2x3+2x3x4+3x4x5x+---
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

1x2x3+2x3x4+3x4x5x
%22%22=%22%22%22%22=%22%22sum%28%28k%5E3%2B3k%5E2%2B2k%29%2Ck=1%2Cmatrix%281%2C3%2C%22%22%2C%22%22%2Cn%29%29%22%22=%22%22sum%28%28k%5E3%29%2Ck=1%2Cmatrix%281%2C3%2C%22%22%2C%22%22%2Cn%29%29%22%22%2B%22%22sum%28%283k%5E2%29%2Ck=1%2Cmatrix%281%2C3%2C%22%22%2C%22%22%2Cn%29%29%22%22%2B%22%22sum%28%282k%5E%22%22%29%2Ck=1%2Cmatrix%281%2C3%2C%22%22%2C%22%22%2Cn%29%29%22%22=%22%22

sum%28%28k%5E3%29%2Ck=1%2Cmatrix%281%2C3%2C%22%22%2C%22%22%2Cn%29%29%22%22%2B%22%223%2Asum%28%28k%5E2%29%2Ck=1%2Cmatrix%281%2C3%2C%22%22%2C%22%22%2Cn%29%29%22%22%2B%22%222%2Asum%28%28k%5E%22%22%29%2Ck=1%2Cmatrix%281%2C3%2C%22%22%2C%22%22%2Cn%29%29%22%22=%22%22%28n%5E2%28n%2B1%29%5E2%29%2F4%29%22%22%2B%22%223%28%28n%28n%2B1%29%282n%2B1%29%29%2F6%29%22%22%2B%22%222%28n%28n%2B1%29%2F2%29%22%22=%22%22
%28n%5E2%28n%2B1%29%5E2%29%2F4%29%22%22%2B%22%22%28n%28n%2B1%29%282n%2B1%29%29%2F2%22%22%2B%22%222n%28n%2B1%29%2F2%22%22=%22%22%28n%5E2%28n%2B1%29%5E2%29%2F4%29%22%22%2B%22%22%282n%28n%2B1%29%282n%2B1%29%29%2F4%22%22%2B%22%224n%28n%2B1%29%2F4%22%22=%22%22%28n%5E2%28n%2B1%29%5E2%2B2n%28n%2B1%29%282n%2B1%29%2B4n%28n%2B1%29%29%2F4%22%22=%22%22

Factor out n(n+1) on top:

%22%22=%22%22%28n%28n%2B1%29%28n%5E2%2Bn%2B4n%2B2%2B4++%29++%29%2F4%22%22=%22%22%28n%28n%2B1%29%28n%5E2%2B5n%2B6++%29++%29%2F4%22%22=%22%22%28n%28n%2B1%29%28n%2B2%29%28n%2B3%29%29%2F4

Edwin