The sum 1 + 1 3 + 2 3 1 + 2 + 1 3 + 2 3 + 3 3 1 + 2 + 3 + . . . + 1 3 + 2 3 + 3 3 + . . . + 15 3 1 + 2 + 3 + . . . + 15 - 1 2 1 + 2 + 3 + . . . + 15 is equal to

Step-by-step solution

Given series is 1 + 1 3 + 2 3 1 + 2 + 1 3 + 2 3 + 3 3 1 + 2 + 3 + . . . + 1 3 + 2 3 + 3 3 + . . . + 15 3 1 + 2 + 3 + . . . + 15 - 1 2 1 + 2 + 3 + . . . + 15 First, we find the sum 1 + 1 3 + 2 3 1 + 2 + 1 3 + 2 3 + 3 3 1 + 2 + 3 + . . . + 1 3 + 2 3 + 3 3 + . . . + 15 3 1 + 2 + 3 + . . . + 15 General term for the series, is T r = 1 3 + 2 3 + 3 3 + . . . + r 3 1 + 2 + 3 + . . . + r Using, the sum 1 + 2 + 3 + . . . + n = n n + 1 2 and 1 3 + 2 3 + 3 3 + . . . + n 3 = n n + 1 2 2 , we get T r = r r + 1 2 2 r r + 1 2 &#86