Geometric Sequence Calculator
Calculate nth term and sum of geometric sequence
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.
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.
| First Term (a) | Ratio (r) | First 5 Terms | Application |
|---|---|---|---|
| 1 | 2 | 1, 2, 4, 8, 16 | Binary numbers, doubling |
| 100 | 1.05 | 100, 105, 110.25, 115.76, 121.55 | 5% compound interest |
| 1000 | 0.5 | 1000, 500, 250, 125, 62.5 | Radioactive half-life |
| 3 | 3 | 3, 9, 27, 81, 243 | Powers of 3 |
| 1 | -2 | 1, -2, 4, -8, 16 | Alternating sequence |
| 0.1 | 10 | 0.1, 1, 10, 100, 1000 | Orders of magnitude |
When r > 1, the sequence grows exponentially. When 0 < r < 1, it decays. When r is negative, terms alternate signs.
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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