3.1.12 Patterns

There is really no good trick to give you a quick answer to most pattern problems (especially the ones on the latter stages of the test). However, it is best to try to think of common things associated between the term number and the term itself. For example, you might want to keep in mind: squares, cubes, factorials, and Fibonacci. Let’s look at some example problems:

Problem: Find the next term of 1, 5, 13, 25, 41, … Solution I: So for this, notice that you are adding to each term 4, 8, 12, 16 respectively. So each time you are incrementing the addition by 4 so, the next term will simply be 16 + 4 added to 41 which is 61. Solution II: Another way of looking at this is to notice that $1 = 1^2 + 0^2$, $5 = 2^2 + 1^2$, $13 = 3^2 + 2^2$, $25 = 4^2 + 3^2$, $41 = 5^2 + 4^2$, so the next term is equal to $6^2 + 5^2 = 61$.

Problem: Find the next term of 0, 7, 26, 63, … Solution: For this one, notice that each term is one less than a cube: $0 = 1^3 - 1$, $7 = 2^3 - 1$, $26 = 3^3 - 1$, $63 = 4^3 - 1$, so the next term would be equal to $5^3 - 1 = 124$.

Here are some more problems to give you good practice with patterns:

Problem Set 3.1.12