jmc

Question

Consider this pattern where the positive, proper fractions with denominator

(n + 1)

are arranged in the

n

th row in a triangular formation. The 1st through 4th rows are shown; each row has one more entry than the previous row. What is the sum of the fractions in the 15th row?

Accepted Answer

The fractions in the n^th row are 1/(n + 1), 2/(n + 1), , n/(n + 1), so their sum is 1 + 2 + + nn + 1.For all n, 1 + 2 + + n = n(n + 1)/2, so 1 + 2 + + nn + 1 = n2.In particular, the sum of the fractions in the 15th row is 152.