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**Triangle Numbers**

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The

**natural numbers**are given by 1, 2, 3, 4, 5, 6, ... etc.1

2

3

4

5

6

The

**triangle numbers**are the sums of the first**n**natural numbers.The

**triangle numbers T**_{n }**are the sums of the first****n**natural numbers.T

_{1}= 1T

_{2}= 1 + 2T

_{3}= 1 + 2 + 3T

_{4}= 1 + 2 + 3 + 4T

_{5}= 1 + 2 + 3 + 4 + 5T

_{6}= 1 + 2 + 3 + 4 + 5 + 6.

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etc.

They are so named because they form a triangle of size n.

The

**tetrahedral numbers**are the sums of the first**n**triangle numbers.**Challenge [hard]:**What comes next?