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An Odd Pyramid (Posted on 2003-10-14) Difficulty: 3 of 5
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.

See The Solution Submitted by DJ    
Rating: 4.1667 (12 votes)

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Solution | Comment 10 of 21 |
In a series of m odd numbers the last number is 2m - 1 and the sum of the numbers is (2m - 1 + 1)m/2 = m^2

In a pyramid of odd numbers of n rows there are (n + 1)n/2 numbers. The sum of the numbers is (n + 1)^2(n^2)/4 = (n^4 + 2n^3 + n^2)/4.

In a pyramid of odd numbers of (n - 1) rows there are n(n - 1)/2 numbers. The sum of the numbers is n^2(n - 1)^2/4 = (n^4 - 2n^3 + n^2)/4.

Therefore, the sum of the numbers in the nth row is 4n^3/4 = n^3

  Posted by retiarius on 2003-10-16 05:13:03
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