Sequences and Progressions
Core concepts
Know the definitions of arithmetic and geometric progressions, their general terms, sums, and convergence conditions. Arithmetic: aₙ = a₁ + (n − 1)d, Sₙ = n(a₁ + aₙ)/2. Geometric: aₙ = a₁qⁿ⁻¹, Sₙ = a₁(qⁿ − 1)/(q − 1) for q ≠ 1, and S = a₁/(1 − q) for |q| < 1. Recognize recursive definitions, monotonicity, and applications such as savings and growth problems.
Traps
- Confusing n with n − 1 in the general term, especially when the first term is given as a₁.
- Using the infinite-sum formula when |q| ≥ 1.
- Sign errors when q is negative and n is even.
- In word problems, miscounting the number of terms (e.g., years from 2010 to 2020 inclusive is 11 terms).
- Mixing up the arithmetic mean (a + c)/2 and the geometric mean √(ac).
Worked example
In an arithmetic progression, a₅ = 17 and a₁₀ = 37. Find a₁ and d. Subtract: 5d = 20, so d = 4. Then a₁ = 17 − 4·4 = 1. The sum of the first 20 terms is S₂₀ = 20/2 · (2·1 + 19·4) = 10 · 78 = 780.
Rule of thumb
Write both given terms using a₁ and d (or q) and subtract to eliminate a₁; this gives the common difference or ratio in one step.