Geometric Sequence Calculator – Find Terms & Sum Online

Calculate any term, common ratio, or sum of a geometric sequence with our free online calculator. Solve geometric progressions for any number of terms with full solutions.

First Term (a₁):

Common Ratio (r):

Number of Terms (n):

Examples:

Understanding Geometric Sequences

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant called the common ratio. If you start with 2 and multiply by 3 each time, you get: 2, 6, 18, 54, 162... This pattern appears everywhere from finance to biology.

Geometric sequences model exponential growth and decay. Population growth, compound interest, radioactive decay, and the spread of viruses all follow geometric patterns. The common ratio determines whether the sequence grows (r > 1), shrinks (0 < r < 1), or alternates (r < 0).

Geometric Sequence Formulas

nth Term Formula

aₙ = a₁ × r^(n-1)

To find any term, multiply the first term by the common ratio raised to the power of (position - 1).

Sum Formula

Sₙ = a₁(1 - r^n) / (1 - r)

The sum of the first n terms. When r = 1, the sum is simply n × a₁.

Worked Examples

Example 1: Find the 8th term of 2, 4, 8, 16...

a₁ = 2, r = 2, n = 8

a₈ = 2 × 2^(8-1) = 2 × 2^7

a₈ = 2 × 128 = 256

The 8th term is 256

Example 2: Sum of first 6 terms of 1, 3, 9, 27...

a₁ = 1, r = 3, n = 6

S₆ = 1 × (1 - 3^6) / (1 - 3)

S₆ = (1 - 729) / (-2)

S₆ = -728 / -2 = 364

Sum = 1 + 3 + 9 + 27 + 81 + 243 = 364 ✓

Example 3: Decay Sequence (r = 0.5)

a₁ = 100, r = 0.5

Sequence: 100, 50, 25, 12.5, 6.25, 3.125...

a₆ = 100 × 0.5^5 = 100 × 0.03125 = 3.125

Each term is half the previous – exponential decay

Example 4: Alternating Sequence (r = -2)

a₁ = 3, r = -2

Sequence: 3, -6, 12, -24, 48, -96...

a₅ = 3 × (-2)^4 = 3 × 16 = 48

Negative ratio creates alternating signs

Quick Fact

The famous "wheat and chessboard" problem involves a geometric sequence. When a inventor asked for 1 grain on the first square, 2 on the second, 4 on the third (doubling each time), the total for all 64 squares is 2^64 - 1 = over 18 quintillion grains – more wheat than exists on Earth!

Frequently Asked Questions

What's the difference between geometric and arithmetic sequences?

Arithmetic sequences add a constant each time (2, 5, 8, 11...). Geometric sequences multiply by a constant each time (2, 6, 18, 54...). Arithmetic grows linearly; geometric grows exponentially.

What happens when the ratio is negative?

The sequence alternates between positive and negative values. For r = -2: a, -2a, 4a, -8a, 16a... The magnitude still grows exponentially, but the sign flips each term.

Can the ratio be a fraction?

Yes! When 0 < r < 1, the sequence decays toward zero. For example, with r = 0.5: 100, 50, 25, 12.5, 6.25... This models radioactive decay and depreciation.

What's an infinite geometric series?

When |r| < 1, the infinite sum converges to a finite value: S∞ = a₁ / (1 - r). For example, 1 + 1/2 + 1/4 + 1/8 + ... = 2. If |r| ≥ 1, the sum diverges to infinity.

How do I find the common ratio?

Divide any term by the previous term: r = aₙ / aₙ₋₁. For the sequence 2, 6, 18, 54: r = 6/2 = 3, or r = 18/6 = 3. The ratio should be the same for any consecutive pair.

Where are geometric sequences used in real life?

Compound interest, population growth, radioactive decay, computer algorithm analysis, fractal geometry, musical scales, and the spread of diseases all involve geometric sequences.

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