| Formula | Expression | Condition |
|---|---|---|
| nth term | a_n = ar^{n-1} | |
| Sum of n terms | S_n = a(r^n - 1)/(r - 1) | r != 1 |
| Sum of n terms | S_n = a(1 - r^n)/(1 - r) | r != 1 |
| Sum of n terms | S_n = na | r = 1 |
| Sum to infinity | S = a/(1-r) | |r| < 1 |
| GP condition | b^2 = ac |
Useful tricks:
- For 3 terms in GP, assume a/r, a, ar (product = a^3)
- For 4 terms in GP, assume a/r^3, a/r, ar, ar^3 (common ratio = r^2)
- Product of n terms of GP: P = (a_1 * a_n)^{n/2} = a^n * r^{n(n-1)/2}
- If each term is multiplied by a constant, the sequence is still GP
- Sum of infinite GP converges iff |r| < 1
- r = (any term) / (preceding term)