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## 1 Adding Two Consecutive Triangular Numbers

One interesting result from triangular numbers is that the sum of two consecutive numbers is a perfect square. In fact, if T n represents the n th triangular number, then T n-1 + T n = n 2.

Example:

Let T n represent the n th triangular number. Find T 7 + T 8 .

Since we are adding the seventh triangular number to the eighth triangular number, we need to compute the square of 8. 8 2 = 64 and thus, T 7 + T 8 = 64. You could verify that T 7 = 28 and T 8 = 36 and then add 28 + 36 = 64.

T 7 + T 8 = 64.

## 2 Adding a Series of Triangular Numbers

To add a series of triangular numbers starting with 1, you need to first find the index of the triangular number is used last. For example, if we were adding 1 + 3 + 6 + 10, then the index of the last number is 4, because we are adding up to the fourth triangular number. The sum of these triangular numbers is given by the formula with the index equal to n:

In fact, these numbers are called tetrahedral numbers.

Example:

1 + 3 + 6 + 10 + 15 + 21 + 28 =

We are adding up to the seventh triangular number. Setting n = 7 and applying the formula above, we get

Therefore, 1 + 3 + 6 + 10 + 15 + 21 + 28 = 84.