The nth term of a GP

Each step multiplies by r, so the second term is ar, the third ar², and the nth term is aₙ = arⁿ⁻¹. The power of r is one less than the position.

In 3, 6, 12, …, a = 3 and r = 2, so a₁₀ = 3 × 2⁹ = 3 × 512 = 1536.

Finding the position of a given term: in 5, 15, 45, …, the term 3645 needs 5 × 3ⁿ⁻¹ = 3645, so 3ⁿ⁻¹ = 729 = 3⁶, n − 1 = 6 and n = 7.

Two known terms fix a G.P. If a₃ = ar² = 12 and a₆ = ar⁵ = 96, dividing gives r³ = 8, so r = 2 and a = 3; then a₁₀ = 1536.

A finite G.P. is a, ar, …, arⁿ⁻¹; the series a + ar + ar² + … is called a geometric series.