A pattern within the perfect squares.

The perfect squares are the numbers that are the result of multiplying a whole number by itself.

On Tuesday, we saw that every perfect square is the sum of two consecutive triangular numbers, such as 21 + 15 = 36 = 6², illustrated below with 21 black asterisks (6+5+4+3+2+1) and 15 red asterisks (1+2+3+4+5).

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Here is another summation method that gives us the perfect squares, once again illustrated with squares made from asterisks, starting with 1² = 1.

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2² = 4 = 1 + 3.

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3² = 9 = 1 + 3 + 5.


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4² = 16 = 1 + 3 + 5 + 7.

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The most direct way to state this is n² is the sum of the first n odd numbers or n² = 1 + 3 + ...  + (2n - 1).

Taking the sum of a pattern of numbers is very common in math, so we took the capital Greek letter sigma as the symbol for summation.  Blogger software doesn't have an easy way for me to write sigma correctly, so I will add a picture.

The big thing at the front that looks like an M lying on its side is the sigma. I also introduced a new variable k, which is the counting variable. The symbols above and below the sigma mean "k starts at 1, and increases by 1 each time until it gets to n, then we stop". The (2k - 1) shows the things we are adding together.

When k = 1, then 2(1) - 1 = 1.
When k = 2, then 2(2) - 1 = 3.
When k = 3, then 2(3) - 1 = 5.
When k = 4, then 2(4) - 1 = 7.
etc. ...

Tomorrow: a pattern concerning squares and the prime numbers of the form 4k + 1.