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Algebra Sequences (as functions)

Geometric sequences

20 practice questions 2 video lessons Theory + worked examples
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  • Explicit & recursive formulas for geometric sequences | High School Math | Khan Academy Watch
  • How to Write Formulas for Geometric Sequences (Recursive & Explicit) Watch
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Theory

A geometric sequence multiplies by a constant common ratio \(r\):

\[a_n=a_1\,r^{\,n-1}.\]

It behaves like an exponential function of \(n\).

Divide consecutive terms to find the ratio \(r\).
A geometric sequence Geometric terms multiply by a constant ratio, curving upward. n ×2 each step
Terms multiply by \(r=2\) each step.
Geometric sequence Geometric sequence Geometric sequence multiply by a common ratio r nth term: aₙ = a₁ · rⁿ⁻¹ like an exponential function
The geometric rule.

nth term:

\[a_n=a_1\,r^{\,n-1}\]
a sub n equals a one times r to the n minus 1
The exponent is \(n-1\).

How to work with geometric sequences

  1. Find the common ratio \(r\).
  2. Identify the first term \(a_1\).
  3. Use \(a_n=a_1 r^{n-1}\).
  4. Substitute the term number \(n\).
Example 1 — Common ratio
Find the common ratio of \(3,6,12,24,\dots\).
Solution

Divide consecutive terms.

\(\dfrac{6}{3}\)\(=\)\(2\)
the common ratio is 2
Example 2 — nth term
Find the \(5\)th term of \(3,6,12,\dots\).
Solution

Use \(a_n=a_1 r^{n-1}\).

\(a_5\)\(=\)\(3\cdot2^{4}=48\)
the fifth term is 48
Example 3 — Explicit rule
Write an explicit rule for \(3,6,12,\dots\).
Solution

Use \(a_1=3,r=2\).

\(a_n\)\(=\)\(3\cdot2^{n-1}\)
a sub n equals 3 times 2 to the n minus 1
Example 4 — Recursive rule
Write a recursive rule for \(3,6,12,\dots\).
Solution

Each term doubles.

\(a_1=3,\)\(a_n=2a_{n-1}\)
a one is 3 and each term doubles

Common pitfalls

The exponent is \(n-1\), not \(n\).
A common ratio is multiplied, not added.
Divide consecutive terms to find \(r\).

Frequently asked questions

What is a geometric sequence?

A sequence with a constant ratio between terms.

What is the nth term formula?

\(a_n=a_1 r^{n-1}\).

How do you find the common ratio?

Divide any term by the one before it.

How is it like an exponential function?

The common ratio acts as the base.

More in Sequences (as functions)
Arithmetic sequences
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