Lesson 4 of 14
The same step every time
Show on a big screenScores run 5, 8, 11, 14, … Each adds 3. What is the 10th score?
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Step 1 of 4
- Positions 1 to 4: 5, 8, 11, 14.
- Every jump is the same: +3.
- Six more same jumps reach position 10.
- Position 10 is 32 — nine jumps in all, not ten.
The rule
An arithmetic progression (AP) adds the same number at every step.
The number added is the common difference, d. The first term is a.
Here tₙ is the term at position n.
tₙ = a + (n − 1) × d
For 5, 8, 11: a = 5 and d = 3. For n = 10, n − 1 = 9. So t₁₀ = 5 + 9 × 3 = 5 + 27 = 32.
The difference can be negative. In 50, 46, 42, … d = 46 − 50 = −4.
Which term is −10?
Take the list 50, 46, 42, … Set tₙ = −10.
50 + (n − 1) × (−4) = −10.
Take 50 from both sides: (n − 1) × (−4) = −60. Divide by −4: n − 1 = 15. So n = 16.
The first term is the start, and reaching it takes no step. So term n has n − 1 steps.
Try the lab to finish this lesson.