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Recognize patterns and generalize

Pattern reasoning involves identifying repeated or changing structure in numerical, tabular, and visual relationships, determining whether a change is additive, multiplicative, or otherwise rule-governed, and using the rule to extend the pattern and predict unknown terms. Generalizations can be expressed in words, tables, graphs, or simple algebraic expressions with variables, while formal proofs and highly abstract, nonlinear, or recursive generalizations are beyond this level.

Detailed Explanation: Recognize patterns and generalize

Look for what changes from one term to the next. Ask:

  • Is the same number being added or subtracted each time?
  • Is the same number being multiplied or divided each time?
  • Can I describe the rule in words or with a variable?

Example: Find the next two terms and the 20th term in the pattern

4, 7, 10, 13, …4,\ 7,\ 10,\ 13,\ \ldots

Step 1: Compare neighboring terms.

7−4=3,10−7=3,13−10=37-4=3,\qquad 10-7=3,\qquad 13-10=3

The pattern adds 33 each time. This is an additive pattern.

Step 2: Extend the pattern.

Add 33 to 1313:

13+3=1613+3=16

Add 33 again:

16+3=1916+3=19

So the next two terms are 16\boxed{16} and 19\boxed{19}.

Step 3: Describe the general rule.

The first term is 44, and each term increases by 33. For term number nn, start with 44 and add 33 for each step after the first:

4+3(n−1)4+3(n-1)

This can also be simplified:

4+3n−3=3n+14+3n-3=3n+1

So the rule is

term n=3n+1\text{term }n=3n+1

Step 4: Use the rule to find the 20th term.

Substitute n=20n=20:

3(20)+1=60+1=613(20)+1=60+1=61

The 20th term is 61\boxed{61}.

Always check that your rule gives the known terms: when n=1n=1, 3(1)+1=43(1)+1=4, and when n=2n=2, 3(2)+1=73(2)+1=7.

Learn by doing: Recognize patterns and generalize

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    Patterning - Rule for Geometric Pattern

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