Geometric Sequence Formula

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

The Formula

an=a1⋅rn−1a_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=2r = 2; the nnth term is 3⋅2n−13 \cdot 2^{n-1}.

Notation

rr = common ratio, a1a_1 = first term, Sn=a1⋅1−rn1−rS_n = a_1 \cdot \frac{1 - r^n}{1 - r} = sum of first nn terms (r≠1r \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 rr.

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

Formal View

A sequence (an)(a_n) is geometric if ∃r∈R∖{0}:an+1an=r\exists r \in \mathbb{R} \setminus \{0\} : \frac{a_{n+1}}{a_n} = r for all n≥1n \geq 1. General term: an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}. Partial sum: Sn=a1⋅1−rn1−rS_n = a_1 \cdot \frac{1 - r^n}{1 - r} for r≠1r \neq 1.

Worked Examples

Example 1

easy
A geometric sequence has a1=5a_1 = 5 and r=2r = 2. Find a8a_8 and S8S_8.

Answer

a8=640a_8 = 640; S8=1275S_8 = 1275

First step

1
Use the geometric sequence formula an=a1⋅rn−1a_n = a_1 \cdot r^{n-1} with a1=5a_1 = 5, r=2r = 2, n=8n = 8.

Full solution

  1. 2
    Find the 8th term: a8=5⋅27=5⋅128=640a_8 = 5 \cdot 2^7 = 5 \cdot 128 = 640
  2. 3
    Apply the partial sum formula: S8=a1⋅1−r81−r=5⋅1−2561−2=5⋅255=1275S_8 = a_1 \cdot \frac{1-r^8}{1-r} = 5 \cdot \frac{1-256}{1-2} = 5 \cdot 255 = 1275
Geometric sequences grow exponentially. The formula an=a1rn−1a_n = a_1 r^{n-1} uses exponent n−1n-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

  • Using a1rna_1 r^{n} instead of a1rn−1a_1 r^{n-1} — the first term is multiplied by rr only (n−1)(n-1) times to reach the nnth.
  • Confusing ratio with difference — divide neighbors for geometric, subtract for arithmetic.
  • Assuming all geometric sequences grow — if ∣r∣<1|r|<1 the terms shrink toward zero (decay).

Why This Formula Matters

Geometric sequences model compounding — interest, population, halving doses — where each step scales the previous, and they underlie exponential functions and geometric series. The defining check, constant ratio, distinguishes them from arithmetic (constant difference) growth and explains why they explode or vanish so fast. Recognizing it by "Do I get the same number every time I divide a term by the one before it?" — rather than by familiar numbers — is what lets a student tell it apart from arithmetic sequence and geometric series and exponential function in a mixed problem set.

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 rr.

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?

rr = common ratio, a1a_1 = first term, Sn=a1⋅1−rn1−rS_n = a_1 \cdot \frac{1 - r^n}{1 - r} = sum of first nn terms (r≠1r \neq 1).

Why is the Geometric Sequence formula important in Math?

Geometric sequences model compounding — interest, population, halving doses — where each step scales the previous, and they underlie exponential functions and geometric series. The defining check, constant ratio, distinguishes them from arithmetic (constant difference) growth and explains why they explode or vanish so fast. Recognizing it by "Do I get the same number every time I divide a term by the one before it?" — rather than by familiar numbers — is what lets a student tell it apart from arithmetic sequence and geometric series and exponential function in a mixed problem set.

What do students get wrong about Geometric Sequence?

The procedure for geometric sequence is the easy part; the trap is using a1rna_1 r^{n} instead of a1rn−1a_1 r^{n-1}. Asking "Do I get the same number every time I divide a term by the one before it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Geometric Sequence formula?

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

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