CalcRef

Sequences & Series · Sequences

Geometric Sequence

an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}

Each term is obtained by multiplying the previous term by a constant ratio r. Converges if |r| < 1.

Worked examples

Find the 6th term of the geometric sequence with a₁ = 3, r = 2.
  1. a₆ = 3 · 2⁵ = 3 · 32 = 96

Answer: 96

Does {(2/3)ⁿ} converge?
  1. |r| = 2/3 < 1, so lim(n→∞) (2/3)ⁿ = 0

Answer: Converges to 0.

Related formulas

Geometric Series

Practice this and 135 more formulas in the CalcRef workspace — quizzes, reference tables, a 16-category unit converter, and an expression evaluator.