Progressions
What this domain tests
Progressions covers arithmetic progressions (AP), geometric progressions (GP), their nth terms and sums, and simple applications. Questions test formula choice and careful counting of terms.
Core rules
- AP: first term a, common difference d = a₂ − a₁. nth term = a + (n−1)d.
- Sum of n AP 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 of an AP with last term l is l − (n−1)d.
- GP: first term a, common ratio r = a₂/a₁. nth term = ar^(n−1).
- Sum of n GP terms: Sₙ = a(rⁿ−1)/(r−1) for r ≠ 1; if r = 1, Sₙ = na.
- Infinite GP sum = a/(1−r) only when |r| < 1.
Worked example
For the AP 7, 10, 13, …, find the 20th term and the sum of the first 20 terms. Here a = 7 and d = 3. a₂₀ = 7 + (20−1)3 = 7 + 57 = 64. S₂₀ = 20/2(7 + 64) = 10 × 71 = 710. Check the pairing: 20 terms form 10 pairs each adding to 71, so 710 is consistent.
Traps
- Using n instead of n−1 in the nth-term formula.
- Putting the last term into the wrong sum formula.
- Forgetting the special case r = 1 in a GP.
- Using the infinite GP formula when |r| ≥ 1.
- Mishandling a negative common difference or ratio.