Using Triangular Numbers:
The Sum of the 1st n Counting Numbers
Consider the triangular numbers. The first few are depicted below with dot arrays:
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Each new triangular number is formed by adding a new column with one more dot than the last column of the preceding triangular number.
That is, if the first triangular number is 1, the second is obtained by adding a new column with 2 dots; the third is obtained from the second triangular number by adding a new column with 3 dots.
Thus we can make the following table:
| Triangular Number |
Number of Dots By Column |
Total Number of Dots |
|---|---|---|
| 1st | 1 | 1 |
| 2nd | 1 + 2 | 3 |
| 3rd | 1 + 2 + 3 | 6 |
| 4th | 1 + 2 + 3 + 4 | 10 |
| 5th | 1 + 2 + 3 + 4 + 5 | 15 |
| . | . | . |
| nth | 1 + 2 + 3 + ... + n | [n X (n + 1)] ÷ 2 |
Thus, the nth triangular number is equal to the sum of the 1st n Counting Numbers.
Use a similar strategy to solve a problem.
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email the author: Bruce Jacobs
Last modified: Thursday, February 6, 2003