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## Patterns & Sequences Worksheets

### Click on the Patterns & Sequences worksheet set you wish to view below.

What are patterns and sequences?

These two topics are different ways of looking at the same thing. Consider the counting numbers, written in a row: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and on and on...

Now, if we label them as either even or odd, we get this pattern: O, E, O, E, O, E, O and on and on as long as we wish. The pattern is "every other number is even and the rest are odd". The sequence is how they are generated, one after the other. So, the sequence would be, "write an O for odd, and then an E for even and continue, switching every time you write another letter."

A pattern tells us the overall picture and a sequence talks about a stepwise way to get that picture.

 How are patterns and sequences used in the real world? Patterns in nature tell us to look for an underlying rule or cause. For instance, in most places where the oceans meet land, there are two high and two low tides a day. The ocean water rises up the beach and recedes in a regular pattern. This is an important pattern, especially if you have a boat. And it was also known by ancient people that the tidal cycle was related to the moon moving in the sky. It wasn't until we understood gravity that we were able to discover the underlying rule - the moon pulls against the ocean waters (and visa versa) and the attraction makes the oceans bulge, causing the tides. Any sequence or regular pattern gives us a clue. Whether it is a pattern in the location of people with a particular disease, or the sequence of solar flares, the regularity tips us off that there is probably an important natural law at work. A basic problem with sequences. In our example with even and odd numbers, is there a rule behind the sequence? Yes. One rule that works is: Start with 2 and label it E. Then, keep adding two's (one at a time) and label every new number you get with an E. Label everything left O. This might seem like a simple problem, but other problems are solved the same way. For instance, what rule explains all the following facts? An even number times an even number gives an even answer. An even number times an odd number gives an even answer. An odd number times an odd number gives an odd answer. Who invented sequences? The basic patterns and sequences of nature (spring, summer, fall, winter...) must have been well known in pre-history. In a mathematical sense, the oldest known use of an infinite sequence was with Archimedes, who lived in Greece in the third century BC. He wrote out the sum of the infinite sequence made by multiplying each previous step by 1/4. It looks like this: 1 + 1/4 + 1/16 + 1/64 + 1/256... = 4/3. An interesting fact about sequences. An interesting sequence can be generated by starting with one and saying out loud the previous term. Here is how it is made. Start with 1. Say, "One one." (This is what you see, the first 1.) Now, write that down next to the first term, 1, 11. Then, read that as "Two ones" and write that down. Your sequence looks like this: 1, 11, 21. If you continue like this, you get the see-and-say sequence: 1, 11, 21, 1211, 111221... The amazing thing is that the series goes on infinitely and no number will repeat more than 3 times in a row, and no number will be bigger than 3. Try it.
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