Triangular numbers

We learn the perfect squares in school.  Here are the first few starting with 1² = 1.

1² = 1
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2² =4
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3² = 9
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4² = 16
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5² = 25
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Another series of numbers base on shape are the triangular numbers. Here are the first five examples starting with 1.

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1 = 1, the 1st triangular number

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1+2 = 3, the 2nd triangular number

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1+2+3 = 6, the 3rd triangular number

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1+2+3+4 = 10, the 4th triangular number


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1+2+3+4+5 = 15, the 5th triangular number

So the n-th triangular number is the sum of the numbers from 1 to n. There are games where these numbers show up, notably the ten pins in bowling and the fifteen balls other than the cue in pool. There are many places where these number show up in math, including the total number of handshakes necessary in a room with n+1 people so that every possible pair of people shakes hands.  Let's look at the example with 5 people.

Person #1: Shakes hands with 4 people
Person #2: Already counted the handshake with #1, 3 people left.
Person #3: Already counted the handshakes with #1 and #2, 2 people left.
Person #4: Already counted the handshakes with #1, #2 and #3, 1 person.
Person #5: Already counted all the handshakes.

There is a formula for the n-th triangular number, which is ½[n(n + 1)].

Consider our idea called relatively prime, which we discussed yesterday.

Two consecutive square numbers are always relatively prime, like 4² and 5². This follows from the fact that any two consecutive numbers are relatively prime.

After the first pair of triangular numbers, namely 1 and 3, any two consecutive triangular numbers cannot be relatively prime.

Ignoring 1 and 3, the next pair are 3 and 6, which have 3 in common as a factor.

Next are 6 and 10, which have 2 in common.

Next, 10 and 15 have 5 in common as a factor.

The formula shows us the numbers are  ½[n(n + 1)] and ½[(n + 1)(n + 2)]. If n is even, the common factor will be n + 1. If n is odd, the common factor will be ½(n + 1).

The square and triangular numbers have a connection. The sum of two consecutive triangular numbers is always a square.

1 + 3 = 4
3 + 6= 9
6 + 10 = 16
10 + 15 = 25
etc. ...

Here is a picture of 10 + 15 = 25 to give an idea why this always works.

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Tomorrow: the Fibonacci numbers.