Geometric Sequence Calculator
Calculate nth terms, sums, and find missing values in geometric sequences with detailed step-by-step solutions and convergence analysis.
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Result
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Geometric Sequence Formulas
nth Term Formula
aₙ = a₁ × r^(n-1)
Where: a₁ = first term, r = common ratio, n = position
Sum Formulas
Sₙ = a₁(1 - r^n)/(1 - r)
S∞ = a₁/(1 - r) if |r| < 1
Finite sum (r ≠ 1) and infinite sum (convergent)
Examples
Example 1: Powers of 2
Sequence: 1, 2, 4, 8, 16, ...
First term (a₁) = 1, Common ratio (r) = 2
5th term: a₅ = 1 × 2^(5-1) = 16
Example 2: Convergent Series
Sequence: 1, 1/2, 1/4, 1/8, 1/16, ...
First term (a₁) = 1, Common ratio (r) = 1/2
Infinite sum: S∞ = 1/(1 - 1/2) = 2
Frequently Asked Questions
A geometric sequence is a sequence where each term after the first is obtained by multiplying the previous term by a constant value called the common ratio. For example: 2, 6, 18, 54... (common ratio = 3).