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Question
Write the first five numbers in the third slanting row of the Pascal’s Triangle and find their squares. What do you infer?
Solution
Numbers in the 3rd standing row are 1, 3, 6, 10, 15, 21, ….
The squares are 12, 32, 62, 102, 152, 212, ….
= 1, 9, 36, 100, 225, 441, …
| Natural Number |
Cubes | Sum of the cubes | Squares of triangular Nos. |
| 1 | 13 = 1 | 1 | 1 |
| 2 | 23 = 8 | 1 + 8 = 9 | 9 |
| 3 | 23 = 27 | 1 + 8 + 27 + 36 | 36 |
| 4 | 43 = 64 | 1 + 8 + 27 + 64 = 100 | 100 |
| 5 | 53 = 125 | 1 + 8 + 27 + 64 + 125 = 225 | 225 |
| 6 | 63 = 216 | 1 + 8 + 27 + 64 + 125 + 216 = 441 | 441 |
| 7 | 73 = 343 | 1 + 8 + 27 + 64 + 125 + 216 + 343 = 784 | 784 |
| . | . | . | . |
| . | . | . | . |
| . | . | . | . |
From the above table we can conclude that the squares of the triangular numbers are the sum of cubes of natural numbers.
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