Triangular Number Know-How…

Triangular Number Know-How

What is a triangular number? It is easy to see what a triangular number looks like, but isn’t there some way of figuring them out without drawing a picture?

Actually, triangular numbers are created by adding consecutive numbers.

For example:  1  =  1

1 + 2  =  3

1 + 2 + 3  =  6

1 + 2 + 3 + 4  =  10

Here’s a challenge for you: What is the 100th triangular number?

ASIDE:  The Great Gauss

How would you go about finding the sum of all of the numbers from 1 to 100? If you don’t feel like spending your whole afternoon with a pencil and paper, you can do what Carl Gauss did to solve this problem in less than a minute.

Carl Gauss, mathematician-extraordinaire,  lived from 1777 to 1855 in what is now Germany. When Gauss was in elementary school, his teacher asked him to find the sum of the first hundred counting numbers. Instead of adding 1 + 2 + 3 + 4… etc., Gauss saw that he could find the answer much quicker if he added pairs of numbers: 1 + 100,  2 + 99,  3 + 98… etc.  When he added the pairs, he always got 101. Since there are 50 total pairs in the numbers from 1 to 100, Gauss simply multiplied 101 x 50 to get 5050. His teacher was astounded! Are you?

Triangular Number Trivia

What can you notice about triangular numbers?

Here are a couple of triangular ticklers.

  • A triangular number never ends in the digits 2, 4, 7 or 9.
  • If you add two consecutive (side by side) triangular numbers, you’ll always get a square number. Square numbers look like this (1, 4, 9, 16):

*                      *       *                          *     *      *                      *     *     *     *

                        *       *                          *      *      *                     *     *     *     *

                                                              *      *     *                      *     *     *     *

                                                                                                      *     *     *     *

Incidentally, square numbers are a great way to memorize some of your multiplication math facts. For instance, above, you can see that 1 x1 =1, 2 x 2 = 2, 3 x 3 = 9,  and 4 x 4 = 16.

  • A prime number is a number that can only be evenly divided by 1 and itself. There is only one prime triangular number: the number 3.
  • A palindrome is a number that can be read the same backwards or forwards. Here are a few triangular numbers that are also palindromes: 1, 3, 55, 595, 3003.
  • There are only 4 triangular numbers that are also Fibonacci numbers.  Can you find them? (Hint: they are all under 60)

(Answer is 1, 3, 21 and 55)

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