Sequences and series
What this domain covers
Sequences and series questions ask you to work with arithmetic and geometric progressions, sum series written in sigma notation, and decide whether an infinite geometric series converges.
Core rules
- Arithmetic: nth term u_n = a + (n − 1)d; sum S_n = n/2 (2a + (n − 1)d) = n/2 (a + l).
- Geometric: u_n = ar^(n−1); S_n = a(1 − r^n)/(1 − r) for r ≠ 1; S∞ = a/(1 − r) only when |r| < 1.
- Convergence: a geometric series converges exactly when |r| < 1; otherwise it diverges (unless a = 0).
- Sigma notation: Σ from r = 1 to n contains exactly n terms; starting at r = 0 gives n + 1 terms.
- Key technique: for a geometric progression, dividing the equation for u₂ by the equation for u₁ eliminates a and yields r.
Worked example
A geometric series has second term 6 and sum to infinity 32. Find r and a. ar = 6 and a/(1 − r) = 32, so a = 32(1 − r). Substituting: 32r(1 − r) = 6, that is 32r² − 32r + 6 = 0, or 16r² − 16r + 3 = 0. Then r = (16 ± √(256 − 192))/32 = (16 ± 8)/32, so r = ¾ with a = 8, or r = ¼ with a = 24. Both satisfy |r| < 1, so both answers are admissible.
Traps
- Using n rather than n − 1 as the power in ar^(n−1).
- Applying S∞ when |r| ≥ 1, where the sum does not exist.
- Discarding the second root of the quadratic, though both may satisfy |r| < 1.
- Miscounting terms when the sigma limits are shifted.