Sequences and Progressions
Core concepts
Distinguish arithmetic (constant difference d) and geometric (constant ratio r) sequences. Know formulas for the n-th term and partial sums, plus infinite geometric series when |r| < 1. Recognize recursive definitions and telescoping patterns.
Traps
- Indexing: a_n = a_1 + (n − 1)d, not nd.
- Sum formulas: S_n = n(a_1 + a_n)/2 for arithmetic; S_n = a_1(1 − rⁿ)/(1 − r) for geometric when r ≠ 1.
- Infinite geometric series converges only when |r| < 1; S = a_1/(1 − r).
- Sign of r: a negative ratio alternates signs; check whether the problem wants a specific sign.
- Number of terms: count carefully when the sequence starts at n = 0 or at some other index.
Rule of thumb
When given two terms, set up a system: for arithmetic, d = (a_m − a_n)/(m − n); for geometric, r^(m−n) = a_m/a_n.
Worked example
An arithmetic sequence has a_3 = 7 and a_8 = 22. Then d = (22 − 7)/(8 − 3) = 3. So a_1 = a_3 − 2d = 7 − 6 = 1. The sum of the first 10 terms is S_10 = 10(1 + a_10)/2, where a_10 = 1 + 9·3 = 28, giving S_10 = 10·29/2 = 145.