Sequence and Series

1 · Formulas and when each holds

• AP: aₙ = a + (n − 1)d; Sₙ = n/2 (2a + (n − 1)d).
• GP: aₙ = arⁿ⁻¹; Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1; S∞ = a/(1 − r) when |r| < 1.
• AM ≥ GM ≥ HM for positive numbers, equal only when all are equal.

2 · Question types and the first step

• Terms given in a GP: write them as a/r, a, ar.
• Sum of a series with a pattern in the differences: find the general term first.

3 · Traps

S∞: only when |r| < 1; check before using it.

4 · My additions

• For three terms in AP, take a − d, a, a + d.