Consider the numerical pyramid below, formed by simply putting down the series of odd numbers into a pyramid.

           1
         3   5
       7   9   11
    13  15  17   19
      . . .

Find a formula for the sum of the numbers in the nth row, and prove it.

The answer is n³

Proof:
The first element in the nth row is given by:
first element = n² - n + 1

since each row is an arithmetic sequence (with a gap of 2), and each row has n elements, the 'middle' element is equal to the first element + (n-1)

This is:
[ (n² - n + 1) + (n - 1) ] = n²

Then, the sum of the whole row is equal to the middle value, multiplied by the number of elements (n)... n² * n = n³

Posted by SilverKnight on 2003-10-14 13:01:16