You've heard of square numbers, or numbers that can be represented by arrays of dots that form squares. The square number tells you the total number of dots in the square, and the side length of the square is the square root of the square number. Other mathematicians, going all the way back to Pythagoras (of Pythagorean Theorem fame), understood this geometric connection to special types of numbers. Pythagoras and his students also studied triangular numbers. Watch the video below for a short study of triangular numbers.

Source: Triangular Numbers, Terry Lindenmuth, YouTube

In the video, you saw a way to generate a triangular number by adding the next counting number to the previous triangular number. That method of generating a value from the value that precedes it in a sequence is called recursion. But how can you identify the value of a triangular number if you only know which number in the sequence of triangular numbers it is? For example, what is the value of the 15th triangular number?

Generating a square number in the sequence is fairly easy – there is an operation for that procedure. If you want to know the value of the 15th square number, you can compute 152 to find that the 15th square number is 152 = 225.

There is also a formula to generate triangular numbers.

Fill in a table like the one shown. You may create the table in your notes.

 Number in Sequence Process to find the Triangular Number Triangular Number 1 1 1 2 1 + 2 3 3 1 +2 + 3 6 4 5 6 n T =______

Let n represent the number in the sequence of triangular numbers, and T represent the triangular number.

1. Is there a constant rate of change in the triangular number? How can you tell?
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2. There are several ways to generate a non-linear function model. In this case, we will try an appropriate quadratic regression to see if that fits the data.

3. Review the table. Describe the pattern.
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The pattern is the number in the sequence times the next number in the sequence all divided by 2. Write a quadratic expression that models the data. You can use your calculator.
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5. How does the formula for generating triangular numbers compare to the formula for finding the area of a triangle?
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Both formulas take two numbers, multiply them, then divide by 2. Use your findings to respond to the prompt below.

In your notes, do the following:

Square numbers can be found using the formula, S = n2, and triangular numbers can be found using the formula,
T = x(x + 1) 2 . Connect each of these formulas to the formulas for finding the area of a square and the area of a triangle, respectively. Based on what you observe, how might you find the value of a pentagonal number?