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Geometric Sequence Calculator

Calculate nth term and sum of geometric sequence

How to Calculate Geometric Sequences

Step 1: Enter the first term (a) of your sequence.

Step 2: Enter the common ratio (r) - the factor each term is multiplied by.

Step 3: Enter how many terms (n) to calculate, then click Calculate.

Understanding Geometric Sequences

What Is a Geometric Sequence

A geometric sequence multiplies each term by a constant called the common ratio. Start with 2 and ratio 3: 2, 6, 18, 54, 162... Each term is 3x the previous one. This is different from arithmetic sequences that add a constant amount.

The Geometric Sequence Formula

Two key formulas:

n-th Term: aₙ = a × r^(n-1)

Find any term directly. For a=2, r=3, the 5th term is 2 × 3⁴ = 2 × 81 = 162.

Sum: Sₙ = a(1-rⁿ)/(1-r)

Sum of first n terms. For a=2, r=3, n=5: S₅ = 2(1-3⁵)/(1-3) = 2(1-243)/(-2) = 242.

Where Geometric Sequences Appear

Compound interest is geometric - your money grows by a fixed percentage each year. Population growth follows geometric patterns. Computer algorithms often have geometric time complexity. Radioactive decay is geometric with ratio less than 1.

Geometric Sequence Examples Reference
First Term (a)Ratio (r)First 5 TermsApplication
121, 2, 4, 8, 16Binary numbers, doubling
1001.05100, 105, 110.25, 115.76, 121.555% compound interest
10000.51000, 500, 250, 125, 62.5Radioactive half-life
333, 9, 27, 81, 243Powers of 3
1-21, -2, 4, -8, 16Alternating sequence
0.1100.1, 1, 10, 100, 1000Orders of magnitude

When r > 1, the sequence grows exponentially. When 0 < r < 1, it decays. When r is negative, terms alternate signs.

Frequently Asked Questions

What is the formula for geometric sequences?
The n-th term is aₙ = a × r^(n-1), where a is the first term and r is the common ratio. The sum of n terms is Sₙ = a(1-rⁿ)/(1-r) when r ≠ 1.
How do you find the common ratio?
Divide any term by the previous term. For 2, 6, 18, 54: ratio = 6/2 = 3, or 18/6 = 3, or 54/18 = 3. The ratio is constant throughout the sequence.
What's the difference between geometric and arithmetic sequences?
Arithmetic sequences add a constant (2, 5, 8, 11... adds 3 each time). Geometric sequences multiply by a constant (2, 6, 18, 54... multiplies by 3). Geometric growth is much faster than arithmetic growth.
Can the common ratio be negative?
Yes. A negative ratio creates an alternating sequence where signs flip each term. For a=1, r=-2: 1, -2, 4, -8, 16, -32... The absolute values still follow the geometric pattern.
What happens when the ratio is less than 1?
The sequence decays toward zero. For a=100, r=0.5: 100, 50, 25, 12.5, 6.25... This models radioactive decay, depreciation, and diminishing returns. The sum converges to a finite value as n approaches infinity.

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