The Sum of the First n Odd Counting Numbers
Consider the following arrays of square numbers:
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In each case the blue dots represent the previous square number and the green dots represent the additional elements of the array that are needed to make the next square number.
Thus, in the third array we have four blue dots which represent the second square number and five green dots which are needed to make the next square.
Notice the pattern: each time we create a new square we add an odd number of dots in sequential order. That is, we begin with 1 dot, then add 3 dots, then add 5 dots, etc. In the process of moving from square number to square number you always add the next odd number of dots.
Consider the following table:
| Number of Odd Nos. Added |
Addends | Sum |
|---|---|---|
| 1 | 1 | 1 = 12 |
| 2 | 1 + 3 | 4 = 22 |
| 3 | 1 + 3 + 5 | 9 = 32 |
| 4 | 1 + 3 + 5 + 7 | 16 = 42 |
| 5 | 1 + 3 + 5 + 7 + 9 | 25 =52 |
| 6 | 1 + 3 + 5 + 7 + 9 + 11 | 36 = 62 |
Therefore, it is clear that:
the sum of the 1st n odd numbers = n2
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email the author: Bruce Jacobs
Last modified: Thursday, February 6, 2003