Arithmetic Progressions
This domain tests recognition of an AP, the nth term, the sum of n terms, and applications such as counting terms and finding unknown terms.
Core rules
- An AP has a constant common difference d = aₙ − aₙ₋₁.
- nth term: aₙ = a + (n − 1)d.
- Sum of n terms: Sₙ = n/2[2a + (n − 1)d] = n/2(a + l), where l is the last term.
- The nth term from the end of a finite AP is l − (n − 1)d.
- Three terms in AP can be taken as a − d, a, a + d; four terms as a − 3d, a − d, a + d, a + 3d.
- To find n when the last term is known, solve a + (n − 1)d = l.
Worked example
The 7th term of an AP is 34 and the 13th term is 64. Find the AP and S₂₀. a + 6d = 34 and a + 12d = 64. Subtracting gives 6d = 30, so d = 5. Then a + 30 = 34, so a = 4. The AP is 4, 9, 14, 19, … S₂₀ = 20/2[2(4) + 19(5)] = 10(8 + 95) = 1030. A quick check: the 13th term is 4 + 12 × 5 = 64, as given.
Traps
- Using n instead of n − 1 in the nth-term formula.
- Miscounting the number of terms when an AP is given as a list.
- Forgetting that d may be negative or fractional.
- Applying Sₙ = n/2(a + l) when l is not actually the last term.