Arithmetic Progressions
Questions cover identifying an AP, finding the nth term and the sum of n terms, and modelling situations such as savings, seating rows and salaries that change by a fixed amount.
Core rules
- An AP has a constant difference d = aₙ − aₙ₋₁.
- 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.
- The nth term from the end is l − (n − 1)d.
- Three terms in AP are conveniently written a − d, a, a + d; four terms as a − 3d, a − d, a + d, a + 3d.
- aₙ = Sₙ − Sₙ₋₁ recovers a term from sums.
Worked example
For the AP 5, 8, 11, …, find the 20th term and the sum of the first 20 terms. Here a = 5 and d = 3. a₂₀ = 5 + 19 × 3 = 62. Then S₂₀ = 20/2(5 + 62) = 10 × 67 = 670.
Traps
Using nd instead of (n − 1)d; substituting n = 20 into n/2(a + l) with the wrong last term; ignoring a negative common difference; misreading 'sum of first n terms' as the nth term; off-by-one errors when counting terms between two given terms.