Finding the Rule of a Pattern (Grades 6–7)
Given a table of positions and values, the goal is a rule that turns the position (n) into the value — usually a straight-line rule like value = a·n + b. Find "a" from how much the value changes each step (the common difference), then find "b" by checking position 1. Position 1→4, 2→7, 3→10 changes by 3 each step (a = 3), and 3·1 + b = 4 gives b = 1, so value = 3n + 1.
Understanding finding the rule of a pattern
Given a table of positions and values, the goal is a rule that turns the position (n) into the value — usually a straight-line rule like value = a·n + b. Find "a" from how much the value changes each step (the common difference), then find "b" by checking position 1. Position 1→4, 2→7, 3→10 changes by 3 each step (a = 3), and 3·1 + b = 4 gives b = 1, so value = 3n + 1.
Key Idea
Given a table of positions and values, the goal is a rule that turns the position (n) into the value — usually a straight-line rule like value = a·n + b. Find "a" from how much the value changes each step (the common difference), then find "b" by checking position 1. Position 1→4, 2→7, 3→10 changes by 3 each step (a = 3), and 3·1 + b = 4 gives b = 1, so value = 3n + 1.
Seeing it in action
Worked example
Find the rule: position 1→4, 2→7, 3→10.
Step (a) = 3. At n = 1: 3·1 + b = 4 → b = 1. → value = 3n + 1.
Use the step to build a straight-line rule.
Try a few
1
5, 2
1
2, 2
1
0, 2
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