Mathematics, Statistics & Geometry

Geometric Sequence Calculator

Use this free Geometric Sequence Calculator to find the common ratio, nth term, or sum of any geometric sequence. View step-by-step formulas.

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 Summary1. Formula nth Term: an = a1 × r^(n-1), Sum: Sn = a1 × (1 - r^n) / (1 - r) 2. Your Inputs First Term (a1) = 2, Ratio (r) = 1.5, Target Term (n) = 5 3. Calculation Steps n-th Term: an = 2 × 1.5^(5-1) = 10.125000 Sum: Sn = 2 × (1 - 7.5938) / (1 - 1.5) = 26.375000 Sum to Infinity: Diverges (Ratio >= 1 or <= -1)

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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

Quick Example: Exponential Growth

If you start with 2 and multiply by 3 each time (2, 6, 18...):

  1. First Term (a₁) is 2.
  2. Common Ratio (r) is 3.
  3. Term Number (n) you want is 5.

Using the formula aₙ = a₁ × r^(n-1), the 5th term is 2 × 3^4 = 162. The sum of all 5 terms is 242.

The Power of Compound Multiplication

Geometric sequences demonstrate exponential growth, which can be unintuitive to the human brain. If you start with a penny ($a_1 = 0.01$) and double it every day ($r = 2$), by day 30 ($n = 30$), 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 numerical pattern where each subsequent term is generated by multiplying the previous term by a fixed, non-zero value known as the common ratio.

Arithmetic progressions rely on constant addition to create linear growth, whereas this type of sequence uses constant multiplication to produce exponential growth.

If the multiplier is a fraction between -1 and 1, the terms progressively shrink toward zero. They eventually become so infinitesimally small that adding an infinite number of them together results in a capped, finite total.

The sequence will oscillate strictly between positive and negative numbers, creating an alternating pattern.

If the absolute value of the multiplier is 1 or greater, the numbers continuously expand. Attempting to sum an infinite amount of these terms causes the total to explode to infinity.

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