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Purcell Math

Math Methods Grades 4-8

Predicting the Next Three Terms of a Sequence

Introduction

Recognizing patterns in sequences is an important skill for predicting the next terms in a series of numbers. Let’s look at some examples and discover how patterns help us find the next terms.

Core Skills

1. Example: Arithmetic Sequence
Sequence: 3,5,7,9,…
This is an arithmetic sequence, where each term increases by +2.
Next three terms:

\[ 9 + 2 = 11, 11 + 2 = 13, 13 + 2 = 15 \]

So, the next three terms are 11,13,15.

2. Example: Repeating Pattern
Sequence: −1,2,6,−1,2,6,…
This sequence repeats every three terms.
Next three terms: Starting again from −1,2,6, we find the next terms are −1,2,6.

3. Example: Square Numbers
Sequence: 1,9,25,49,…
Each term is a perfect square: 1²,3²,5²,7²
Next three terms:

\[ 9^2 = 81, 11^2 = 121, 13^2 = 169 \]

So, the next terms are 81, 121, 169.

4. Example: Cube Numbers
Sequence: 1, 8, 27, 64,…
This is a sequence of cube numbers: 1³,2³,3³,4³.
Next three terms:

\[ 5^3 = 125, 6^3 = 216, 7^3 = 343 \]

So, the next terms are 125,216,343.

5. Example: Simple Counting Sequence
Sequence: 7,8,9,…
This sequence simply counts up by 1.
Next three terms:

\[ 10, 11, 12 \]

So, the next three terms are 10,11,1210, 11, 1210,11,12.

Conclusion

Learning to identify different patterns helps in predicting future terms in sequences. Arithmetic sequences, repeating patterns, squares, and cubes are just a few common types. Identifying these can make sequence prediction much easier.

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