Sequences and Series
This domain tests the nth term and sums of arithmetic and geometric progressions, means, and the standard sigma formulae.
Arithmetic progression
With first term a and common difference d: tₙ = a + (n−1)d, Sₙ = (n/2)[2a + (n−1)d] = (n/2)(a + l). The arithmetic mean of a and b is (a+b)/2. If three terms are in AP, write them a−d, a, a+d.
Geometric progression
With first term a and ratio r: tₙ = ar^(n−1), Sₙ = a(rⁿ−1)/(r−1) for r ≠ 1, and S∞ = a/(1−r) only when |r| < 1. Geometric mean of a and b is √(ab).
Standard sums
Σn = n(n+1)/2, Σn² = n(n+1)(2n+1)/6, Σn³ = [n(n+1)/2]².
Worked example
The 5th term of an AP is 17 and the 12th term is 45. Subtracting: (12−5)d = 45 − 17, so 7d = 28, d = 4. Then a = 17 − 4(4) = 1. S₂₀ = (20/2)[2(1) + 19(4)] = 10(2 + 76) = 780. Note the check: the 12th term is 1 + 11(4) = 45, consistent.
Traps that cost marks
- Using the formula for Sₙ when the question asks for tₙ, or vice versa (Sₙ − Sₙ₋₁ = tₙ).
- Applying S∞ = a/(1−r) when |r| ≥ 1; the series then diverges.
- Ignoring the special case r = 1, where Sₙ = na.
- Sign errors when d is negative or r is a fraction; keep the bracket (2a + (n−1)d) intact.