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).

*** * * * * ***

*** * * * * ***

*** * * * * ***

*** * * * * ***

*** * * * * ***

*** * * * * ***

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

*****

2² = 4 = 1 + 3.

*** ***

*** ***

3² = 9 = 1 + 3 + 5.

*** * ***

*** * ***

*** * ***

4² = 16 = 1 + 3 + 5 + 7.

*** * * ***

*** * * ***

*** * * ***

*** * * ***

The most direct way to state this is

*n*² is the sum of the first

*n*odd numbers or

*n*² = 1 + 3 + ... + (2

*n*- 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 (2

*k*- 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 4

*k*+ 1.

## No comments:

## Post a Comment