Geometric Sequence Formulas – Complete Reference

A geometric sequence is a list 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 (r). For example, 2, 4, 8, 16 is a geometric sequence with first term a₁ = 2 and common ratio r = 2. Understanding the formulas that describe such sequences is essential for everything from algebra homework to financial planning. For a refresher on the definition and examples, check out What Is a Geometric Sequence? Definition & Examples (2026).

The nth Term Formula

The most fundamental formula for a geometric sequence gives you any term in the sequence directly:

aₙ = a₁ × r^(n-1)

Here:

• aₙ is the term you want to find (the nth term).
• a₁ is the first term of the sequence.
• r is the common ratio.
• n is the position of the term (e.g., n=1 gives a₁, n=2 gives a₂, etc.).

The exponent n-1 appears because the first term (a₁) has been multiplied by r zero times to get to itself. Each step forward multiplies by r one more time. For instance, the 5th term is a₁ × r⁴. This formula works for any real r (including negative and fractional values) and any positive integer n. To see this calculated step by step, visit the How to Calculate Geometric Sequences: Step-by-Step Guide (2026).

Sum of the First n Terms

Often you need the total of the first n terms in a geometric sequence. The formula depends on whether r equals 1.

When r ≠ 1

Sₙ = a₁ × (1 - rⁿ) / (1 - r)

Breaking it down:

• Sₙ is the sum of the first n terms.
• a₁ is the first term.
• r is the common ratio.
• n is the number of terms.

Why does this work? Write out the series: S = a₁ + a₁r + a₁r² + ... + a₁r^(n-1). Multiply both sides by r: rS = a₁r + a₁r² + ... + a₁rⁿ. Subtract rS from S: most terms cancel, leaving S - rS = a₁ - a₁rⁿ. Factor to get S(1 - r) = a₁(1 - rⁿ), then divide by (1 - r).

When r = 1

If r = 1, every term equals a₁, so the sum of n terms is simply Sₙ = n × a₁.

Sum to Infinity

If a geometric sequence continues forever, you can sometimes add up all its terms and get a finite number. This only happens when the absolute value of the common ratio is less than 1 (|r| < 1). The formula is:

S∞ = a₁ / (1 - r)

Where:

• S∞ is the sum of all terms to infinity.
• a₁ is the first term.
• r is the common ratio (must satisfy |r| < 1).

Intuitively, as n grows large, rⁿ approaches zero, so the sum formula Sₙ = a₁(1 - rⁿ)/(1 - r) becomes a₁/(1 - r). For example, the series 1 + 0.5 + 0.25 + 0.125 + ... has a₁=1, r=0.5, and sums to 2. This idea is used in finance, physics, and many other fields. Learn more in Geometric Sequences in Finance: Compound Interest Applications (2026).

Practical Implications and Edge Cases

Geometric sequences model real-world growth and decay. When |r| > 1, the sequence grows without bound (exponential growth). When |r| < 1, it approaches zero (exponential decay). Two special cases are worth noting:

• r = 0: After the first term, all subsequent terms are zero. The formula still works.
• r = -1: The sequence alternates between a₁ and -a₁. The sum of the first n terms is either a₁ (if n is odd) or 0 (if n is even). The sum to infinity does not exist because the terms do not settle.
• r = 1: All terms are equal; the sum is just n × a₁. The sum to infinity diverges (unless a₁=0).

When using the sum to infinity formula, always check that |r| < 1. If |r| >= 1, the series diverges and a finite sum does not exist. For a deeper look at what different ratio values mean, see Geometric Sequence Ranges: What Different Values Mean (2026).

Historical Note

Geometric sequences have been studied for thousands of years. Ancient Babylonians and Egyptians used them for practical problems, and the Greek mathematician Euclid wrote about them around 300 BCE. The modern formulas we use today were developed over centuries, with the sum formula emerging in the 17th century.

Whether you're a student or professional, our Geometric Sequence Calculator can handle these calculations instantly. For more common questions, visit the Geometric Sequence FAQs: Common Questions Answered (2026).