Overview:
One of the problem-solving strategies that is often used in math is to look for a pattern. Often when exploring problems, the student can notice a relationship between numbers. This relationship can help to solve the problem by shortening the number of steps it takes to get to a solution.
Step 1: Make a Table
The first step to look for a pattern is to make a table showing the relationships that are there. For example, suppose the problem were to find the next three numbers in the series 2, 4, 6, 8. They are related in a simple way, such that 2 +2 is 4 +2 is 6 +2 is 8. In order to extend the pattern, 8 +2 is 10, +2 is 12 +2 is 14. Therefore, the next three numbers are 10, 12, and 14.
Step 2: Find the Relationship Between Numbers
In the previous example, the relationship was very simple, as the pattern was the set of the first 7 even numbers. Each number in the series was separated by the same distance. Suppose the relationship weren’t quite as simple. This time, the numbers in the sequence are 1, 5, 14, 30, and 55. What are the next 3 numbers in the series? 1 +4 is 5 +9 is 14 +16 is 30 +25 is 55. The numbers aren’t even the same distance from one another.
Step 3: Make a Prediction
What do these numbers have in common? After finding a pattern, it’s possible to predict what the next numbers will be. Those differences 1, 4, 9, 16, and 25, have a pattern in themselves. They are the first 5 squared numbers, as 12 is 1, 22 is 4, 32 is 9, 42 is 16, and 52 is 25. So the next difference might be 62 or 36, and 55 + 36 = 91. The next squared number will be 72, or 49, and 91 +49 = 140. The next squared number will be 82 or 64, and 140 + 64 = 204.
Step 4: Check the Answer
The sequence with the next three numbers is 1, 5, 14, 30, 55, 91, 140, and 204. The pattern fits in every step of the sequence. If the numbers were added carefully, it would be easy to check each one, as 1 + 4 is 5, 5 + 9 is 14, 14 + 16 is 30, 30 + 25 is 55, 55 + 36 is 91, 91 + 49 is 140, and 140 + 64 is 204. The next squared number would be 81 (92), and 204 + 81 is 285.
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