Patterns and Sequence and n! Oh my!!

  Ok, I know what you're thinking: What in the world does The Wizard of Oz have to do with math?!  Well, it is simple: math is just as scary as lions and tigers and bears!  All joking aside, patterns and sequences can be pretty tough sometimes.  Yeah the simple █▲█▲█ is easy enough; however, throw in a couple parallelograms, hexagons, and the occasional potato and then you're really stuck!  Fortunately, I have just learned a tip to these sequences that can help you out!

The Power of n

  When looking at a pattern, it is always helpful to put each term number as n; by coming up with simple formulas, we can figure out random terms in the sequence without having to draw out each pattern consecutively.  So, lets work on an example:

  So the pattern here is pretty simple: with each odd numbered term we add a triangle and with each even numbered term we add 2 squares.  So, this is where it gets complicated: We want to find out how many triangles and squares there are in the 25th and 30th terms, without having to set up all 30 sequences.  So let's take a deep breath and calm down. Remember n?  Let's call him up really quick, because I have a feeling he will help us a great deal!  Let's look at the figures, starting with the triangles.

Term Number	1	2	3	4	5
Number of Triangles	1	1	2	2	3

  I realize by looking at this chart each even numbered term has half the number of triangles.  So, let's put this into a formula using n.  At an even numbered term n, the number of triangles is n÷2. Next, look at the odd numbered terms.  I see that each odd term has the same number of triangles as the following even term.  So, for this formula we are using the same one we just created, except we have to add 1 to make it fit the odd terms.  At an odd number term n, the number of triangles is n+1.
                                                                    2
Now let's take a look at the squares.

Term Number	1	2	3	4	5
Number of Squares	0	2	2	4	4

  I immediately notice that at each even numbered term, the term number is the same as the amount of squares.  So this formula is simple.  At an even numbered term n, the number of squares is n.  I also notice that each odd numbered term has the same number of squares as the term before.  So, we just have to subtract 1 from the formula.  At an odd numbered term n, the number of squares is n-1.  We now have enough information to solve the problem.


1. How many triangles and squares will there be in the 25th sequence?
  25 is an odd number, so we will use the formulas that we created for odd numbers.
  At an odd number term n, the number of triangles is n+1.

                                                                                     2

               n=25          n+1 --> 25+1 --> 26 = 13

                                  2           2          2

 At an odd numbered term n, the number of squares is n-1.
              
               n=25        n-1 --> 25-1 = 24


       Answer:  So, the answer is that in the 25th sequence, there will be 13 triangles and 24 squares.


2. How many triangles and squares will there be in the 30th sequence?
  30 is an even number, so we will use the formulas that we created for even numbers.
  At an even numbered term n, the number of triangles is n÷2.


                n=30       n÷2 --> 30÷2 = 15


 At an even numbered term n, the number of squares is n.


               n=30        30 = 30


       Answer: So, the answer is that in the 30th sequence, there will be 15 triangles and 30 squares.


  As long as it is a pattern of sequence, this tip can be applied!  Well, I'm off to see the wizard, so until next time!