Algebra I to AP Precalculus · Module 10 of 14
Sequences & Data Analysis
Arithmetic and geometric sequences, and reading trends in scatter plot data.
Arithmetic Sequences
- An arithmetic sequence has a constant difference d between consecutive terms; find d by subtracting any term from the one after it.
- The nth term formula is aₙ = a₁ + (n − 1)d, where a₁ is the first term.
Find the 10th term of the sequence 4, 7, 10, 13, ...
Solution. a₁ = 4, d = 3 (each term is 3 more than the last). a₁₀ = 4 + (10 − 1)(3) = 4 + 27 = 31.
Geometric Sequences
- A geometric sequence has a constant ratio r between consecutive terms; find r by dividing any term by the one before it.
- The nth term formula is aₙ = a₁ · r^(n−1).
Find the 6th term of the sequence 3, 6, 12, 24, ...
Solution. a₁ = 3, r = 2 (each term doubles). a₆ = 3 · 2⁵ = 3 · 32 = 96.
Writing & Using Sequence Formulas
- Given any two terms of an arithmetic sequence, subtract the two versions of the nth-term formula to solve for d, then back-substitute to find a₁.
- Arithmetic sequences model constant additive growth (e.g. saving a fixed amount each week); geometric sequences model constant multiplicative growth (e.g. compounding).
An arithmetic sequence has a₃ = 11 and a₇ = 27. Find d and a₁.
Solution. a₇ − a₃ = (7 − 3)d = 4d = 27 − 11 = 16, so d = 4. Then a₃ = a₁ + 2d = 11 → a₁ = 11 − 8 = 3.
Scatter Plots & Line of Best Fit
- A scatter plot shows the relationship between two variables: a positive association means both tend to increase together, a negative association means one tends to increase as the other decreases.
- A line of best fit approximates the overall trend, but a strong association never proves that one variable causes the other — a third, unseen factor could be driving both.
A scatter plot of hours studied vs. test score trends upward. What can and can't you conclude?
Solution. You can conclude a positive association: more hours studied tends to go with a higher score. You can't conclude studying causes the higher score from the correlation alone — other factors (prior knowledge, motivation) could contribute to both.
Common mistake: Correlation is not causation. A strong trend in a scatter plot describes a pattern in the data, not a proven cause-and-effect relationship.
Check yourself
2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.
1. Find the 8th term of the arithmetic sequence 5, 9, 13, 17, ...
2. Find the 5th term of the geometric sequence 2, 6, 18, 54, ...