|
The sum of the terms in the nth row is n3
First, some things to note:
The kth odd number is (2k - 1).
The nth row in this pyramid contains n terms.
After (n-1) rows have been written down, the number of terms that have been written is:
1 + 2 + 3 + ... + n-1 = n(n-1)/2 *
So, the first term of the nth row is the [n(n-1)/2 + 1]th odd number, equal to n(n-1) + 1.
Similarly, the last term of the nth row is the [n(n+1)/2]th odd number, which is n(n+1) - 1.
The terms within a row are an arithmetic series, obviously, and the mean of the terms is equal to the mean of the first and last terms, or:
[n(n-1) + 1 + n(n+1) -1]/2
= (n² - n + n² + n)/2
= 2n²/2
= n²
Thus, the sum of the row can be found by multiplying the number of terms (n) by the average value of the terms (n²), giving the sum of terms in the nth row to be n³. |
