Triangular Numbers & Fibonacci Numbers

dkmathstats (61)in #mathematics • 6 years ago

Hi there. This patterning math post features triangular numbers and Fibonacci numbers.

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Sections

Triangular Numbers
Fibonacci Numbers
The Golden Ratio From Fibonacci Numbers

Triangular Numbers

To start, here is an image which features triangular numbers.

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The first number is 1, the second number is 3 where the number 2 is added to 1. Adding 3 to the second number gives 6 and adding 4 to 6 gives the fourth triangular number. The pattern continues where the number that is added increases by one.

As you can see triangular numbers are based on the sequence:

1, 2, 3, 4, 5, 6, 7, ...

Finding The N-th Triangular Number Formula

Let the variable N represents the N-th triangular number. To find the N-th triangular number you can use the formula:

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Example One

The seventh triangular number can be found with N = 7 in the formula.

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Example Two

The hundredth triangular number can be found with N = 100 in the formula.

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Fibonacci Numbers

There is a neat pattern where the next number in a sequence of numbers depends on the last two numbers. This Fibonacci sequence of numbers is as follows:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

In the number sequence we start with the numbers 1 and 1 as the first two terms. The next number in the sequence consists of adding the last two numbers together.

A more mathematical version would be as follows:

The Golden Ratio Approximation From Fibonacci Numbers

The golden ratio value of can be approximated from the ratio of Fibonacci numbers. This ratio consists of two consecutive numbers where the second number is divided by the first number.