Arithmetic Progressions and Sequences
Arithmetic progression questions in the optional paper are formula-driven: identifying a and d, finding a required term, counting terms, and computing sums. The examiner is checking whether you read the question as “term” or “sum”.
Core rules
- General AP: a, a + d, a + 2d, …, with nth term aₙ = a + (n − 1)d.
- Sum of n terms: Sₙ = n/2 [2a + (n − 1)d], or Sₙ = n/2 (a + l) when the last term l is known.
- aₙ = Sₙ − Sₙ₋₁ for n ≥ 2.
- If a, b, c are in AP then 2b = a + c; the arithmetic mean of a and c is (a + c)/2.
- Sum of first n natural numbers = n(n + 1)/2; sum of the first n odd numbers = n².
- Three convenient unknowns in AP: a − d, a, a + d; four: a − 3d, a − d, a + d, a + 3d.
Worked example
For the AP 7, 10, 13, … (i) which term is 151, and (ii) what is the sum of the first 20 terms? Here a = 7, d = 3. Set 7 + (n − 1)3 = 151, so 3(n − 1) = 144 and n = 49; the 49th term is 151. S₂₀ = 20/2 [2(7) + 19(3)] = 10(14 + 57) = 710.
Traps
- A fractional or negative n means no such term exists; do not round it and carry on.
- Using n instead of (n − 1) in the nth-term formula, which shifts every answer by one term.
- Confusing aₙ with Sₙ, especially when the question says “the sum is 151”.
- Wrong sign for d in a decreasing AP such as 50, 46, 42, …
- Using Sₙ = n/2 (a + l) when l is not actually the last term.