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Sum the rows of a triangular labelled board

The n-th row from the top of a board has n tiles, labelled 1,2,…,n1,2,\ldots,n from left to right, as in the diagram. Given ∑r=1nr2=n(n+1)(2n+1)/6\sum_{r=1}^nr^2=n(n+1)(2n+1)/6, find the sum in the k-th row and hence show the sum of all numbers in n rows is n(n+1)(n+2)/6n(n+1)(n+2)/6. marks

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Given triangular tile board: first row is 1; second row 1,2; third row 1,2,3; ellipses continue the pattern. Every row starts again at 1.
Given triangular tile board: first row is 1; second row 1,2; third row 1,2,3; ellipses continue the pattern. Every row starts again at 1.
Given triangular tile board: first row is 1; second row 1,2; third row 1,2,3; ellipses continue the pattern. Every row starts again at 1.

[2017/NYJC/I/1]

CJC · Review problems · Question 10

Sum the labels within a row first, then sum over the row index.

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