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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?
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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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