Sequences and Series
What the exam tests
Questions test recognition of arithmetic and geometric progressions, computation of the nth term and sums, and convergence of infinite geometric series. Business applications include simple growth and depreciation modelled by AP or GP.
Core rules
AP: tₙ = a + (n−1)d; Sₙ = n/2[2a + (n−1)d] = n/2(a + l). GP: tₙ = ar^(n−1); Sₙ = a(rⁿ−1)/(r−1) for r ≠ 1; S∞ = a/(1−r) only when |r| < 1.
- Three terms in AP: a−d, a, a+d. Three terms in GP: a/r, a, ar.
- Sum of first n natural numbers = n(n+1)/2; squares = n(n+1)(2n+1)/6; cubes = [n(n+1)/2]².
Worked example
The 3rd term of a GP is 12 and the 6th term is 96. Find a and r. t₃ = ar² = 12 and t₆ = ar⁵ = 96. Dividing: r³ = 8, so r = 2. Then a(4) = 12, so a = 3. The 8th term is ar⁷ = 3 × 128 = 384. The sum of the first 8 terms is 3(2⁸−1)/(2−1) = 3(255) = 765.
Traps
- Using S∞ when |r| ≥ 1; the series then diverges (for r = 1 use Sₙ = na).
- Confusing tₙ with Sₙ, or using n instead of n−1 in the nth-term formula.
- Losing the negative sign when r is negative and n is odd or even.
- In word problems, identifying the first term as the value after one year rather than the starting value.