Mathematics, Statistics & Geometry

Geometric Sequence Calculator

Calculate the Nth term, finite sum, and infinite sum of any geometric sequence. Provides exact mathematical formulas for exponential growth.

5th Term (a_n)
10.125
Sum of first 5 terms (S_n)26.375
Sum to Infinity (S_∞)Divergent (r ≥ 1 or r ≤ -1)
Calculation StepsFirst term a₁ = 2, Common ratio r = 1.5, n = 5 Nth Term Formula: a_n = a₁ * r^(n-1) a_5 = 2 * 1.5^(5-1) = 2 * 5.0625 = 10.125000 Sum Formula: S_n = a₁ * (1 - rⁿ) / (1 - r) = 2 * (1 - 1.5^5) / (1 - 1.5) = 26.375000

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Mastering Exponential Growth

The Geometric Sequence Calculator is a critical tool for understanding compound interest, population dynamics, and exponential patterns. It instantly generates the Nth term, finite sums, and tests for infinite series convergence.

an=a1rn−1andSn=a11−rn1−r\begin{aligned} a_n = a_1 r^{n-1} \quad \text{and} \quad S_n = a_1 \frac{1 - r^n}{1 - r} \end{aligned}

Where:
ana_n=
The value of the sequence at position n
a1a_1=
The starting value of the sequence
r=
The multiplier between each term

The Power of Compound Multiplication

Geometric sequences demonstrate exponential growth, which can be unintuitive to the human brain. If you start with a penny (a1=0.01a_1 = 0.01) and double it every day (r=2r = 2), by day 30 (n=30n = 30), you will have over 5 million dollars. This calculator makes that math transparent.

Real-World Applications

  • Finance: Calculating compound interest, mortgage amortizations, and the present value of future annuities.
  • Biology: Modeling the unrestricted reproduction rates of bacteria or virus transmission (the 'R0' value is effectively a common ratio).
  • Computer Science: Analyzing the time complexity of "divide and conquer" algorithms, like Merge Sort, using geometric sums.

Frequently Asked Questions

It 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'.

In an arithmetic sequence, you ADD the same number every time (linear growth). In a geometric sequence, you MULTIPLY by the same number every time (exponential growth).

If the common ratio (r) is a fraction between -1 and 1, the sequence numbers get smaller and smaller. Eventually, they become so small that adding an infinite number of them together still results in a finite sum.

If the ratio is negative, the sequence will alternate between positive and negative numbers. This is called an alternating geometric sequence.

Yes. If the absolute value of the ratio is 1 or greater, the numbers keep growing. If you try to add an infinite amount of them, the sum explodes to infinity (it diverges).

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