Geometric Sequence Formula

The Formula

a_n = a_1 \cdot r^{n-1}

When to use: Multiply by the same number each step — 2, 6, 18, 54, ... (multiply by 3). This is exponential growth.

Quick Example

3, 6, 12, 24, 48, ... — common ratio r = 2; the nth term is 3 \cdot 2^{n-1}.

Notation

r = common ratio, a_1 = first term, S_n = a_1 \cdot \frac{1 - r^n}{1 - r} = sum of first n terms (r \neq 1).

What This Formula Means

A sequence where each term is obtained from the previous by multiplying by a fixed non-zero constant called the common ratio r.

Multiply by the same number each step — 2, 6, 18, 54, ... (multiply by 3). This is exponential growth.

Formal View

A sequence (a_n) is geometric if \exists r \in \mathbb{R} \setminus \{0\} : \frac{a_{n+1}}{a_n} = r for all n \geq 1. General term: a_n = a_1 \cdot r^{n-1}. Partial sum: S_n = a_1 \cdot \frac{1 - r^n}{1 - r} for r \neq 1.

Worked Examples

Example 1

easy
A geometric sequence has a_1 = 5 and r = 2. Find a_8 and S_8.

Solution

  1. 1
    Use the geometric sequence formula a_n = a_1 \cdot r^{n-1} with a_1 = 5, r = 2, n = 8.
  2. 2
    Find the 8th term: a_8 = 5 \cdot 2^7 = 5 \cdot 128 = 640
  3. 3
    Apply the partial sum formula: S_8 = a_1 \cdot \frac{1-r^8}{1-r} = 5 \cdot \frac{1-256}{1-2} = 5 \cdot 255 = 1275

Answer

a_8 = 640; S_8 = 1275
Geometric sequences grow exponentially. The formula a_n = a_1 r^{n-1} uses exponent n-1 because the first term has no multiplications applied.

Example 2

medium
Bacteria double every 3 hours. Starting with 500, how many after 15 hours?

Example 3

medium
Find the 6th term of the geometric sequence: 2, 6, 18, ...

Common Mistakes

  • Confusing the common ratio r with the common difference d: in a geometric sequence you multiply by r, so r = \frac{a_{n+1}}{a_n}, not a_{n+1} - a_n.
  • Using r^n instead of r^{n-1} in the formula: a_n = a_1 \cdot r^{n-1}, not a_1 \cdot r^n — the first term has exponent 0, not 1.
  • Forgetting that a negative ratio makes terms alternate in sign: a_1 = 2, r = -3 gives 2, -6, 18, -54, ... — the absolute values grow but signs alternate.

Why This Formula Matters

Models compound interest, population growth, radioactive decay.

Frequently Asked Questions

What is the Geometric Sequence formula?

A sequence where each term is obtained from the previous by multiplying by a fixed non-zero constant called the common ratio r.

How do you use the Geometric Sequence formula?

Multiply by the same number each step — 2, 6, 18, 54, ... (multiply by 3). This is exponential growth.

What do the symbols mean in the Geometric Sequence formula?

r = common ratio, a_1 = first term, S_n = a_1 \cdot \frac{1 - r^n}{1 - r} = sum of first n terms (r \neq 1).

Why is the Geometric Sequence formula important in Math?

Models compound interest, population growth, radioactive decay.

What do students get wrong about Geometric Sequence?

If |r| < 1, terms shrink toward zero. If |r| > 1, terms grow without bound.

What should I learn before the Geometric Sequence formula?

Before studying the Geometric Sequence formula, you should understand: sequence, exponents.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Growing Patterns, Arithmetic and Geometric Sequences →
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