A Pascal-like triangle has as the top row and followed by as the second row. In each subsequent row the first number is , the last number is , and, as in the standard Pascal Triangle, each other number in the row is the sum of the two numbers directly above it. The first four rows are shown below.
What is the sum of the digits of the sum of the numbers in the th row?
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We are given a Pascal-like triangle whose first rows are
Each row starts with , ends with , and each interior entry is the sum of the two numbers directly above it.
Let denote the sum of the numbers in the -th row (with the top row being ). Then
For , the -th row consists of:
Each interior entry is the sum of two entries from the -st row. When we sum all interior entries, every entry from the -st row is counted twice. We also notice that the first and the last are also being added twice once to calculate the first and last terms respectively and once for calculating the second and the second last terms respectively.
Therefore, when
Since, , we get
The sum of the digits of is
The problems on this page are the property of the MAA's American Mathematics Competitions