Geometric Sequence Calculator

Calculate terms, partial sums, and infinite sums of geometric sequences. Input the first term, common ratio, and number of terms.

Enter Sequence Parameters

Sequence Terms

3, 6, 12, 24, 48, 96, 192, 384

Result

nth Term (an)
384
First Term (a1) 3
Common Ratio (r) 2
Number of Terms (n) 8
Partial Sum (Sn) 765
Infinite Sum (Sinf) Diverges (|r| >= 1)
Sequence Type Divergent

Step-by-Step Solution

an = 3 x 2^(8-1) = 3 x 128 = 384

Understanding Geometric Sequences

A geometric sequence (also called a geometric progression) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. If the first term is a1 and the common ratio is r, the sequence is: a1, a1r, a1r2, a1r3, and so on.

Key Formulas

nth Term

Find any term in the sequence given the first term and ratio.

an = a1 x r^(n-1)

Partial Sum

Sum of the first n terms when r is not equal to 1.

Sn = a1(1 - r^n) / (1 - r)

Infinite Sum

Sum of all terms when |r| < 1 (converging series).

S = a1 / (1 - r)

Common Ratio

The ratio between consecutive terms is constant.

r = a(n+1) / an

Types of Geometric Sequences

  • Convergent (|r| < 1): Terms approach zero. The infinite sum converges to a finite value.
  • Divergent (|r| > 1): Terms grow without bound. The infinite sum does not exist.
  • Alternating (r < 0): Terms alternate between positive and negative.
  • Constant (r = 1): All terms are equal. The sum is simply n times a1.
  • Oscillating (r = -1): Terms alternate between a1 and -a1.

Real-World Applications

Geometric sequences appear throughout mathematics, science, and finance:

  • Compound interest: Account balances grow geometrically when interest compounds at a fixed rate.
  • Population growth: Organisms reproducing at a constant rate form geometric sequences.
  • Radioactive decay: Remaining material follows a geometric decrease over equal time intervals.
  • Computer science: Binary trees, divide-and-conquer algorithms, and data structure analysis involve geometric sums.
  • Music: Frequency ratios between musical notes follow geometric patterns.

Examples

Example 1: 2, 6, 18, 54, ... Here a1 = 2, r = 3. The 10th term is 2 x 39 = 39,366.

Example 2: 100, 50, 25, 12.5, ... Here a1 = 100, r = 0.5. Since |r| < 1, the infinite sum = 100 / (1 - 0.5) = 200.