Geometric Sequences

In a geometric sequence, each term is produced by multiplying the previous term by the same factor.

Constant common ratio

A sequence is geometric when the ratio of consecutive nonzero terms is constant:

For $3,6,12,24,\ldots$, each term is multiplied by $2$, so $r=2$. For $80,40,20,10,\ldots$, $r=1/2$.

If zeros occur, inspect the generating rule carefully because division by zero cannot calculate a ratio.

Recursive form

A geometric sequence can be defined recursively by

where $a$ is the first term and $r$ is the common ratio.

For $5,-10,20,-40,\ldots$,

The negative ratio makes signs alternate.

Explicit form

From the first term to the $n$th term, the ratio is applied $n-1$ times:

For $3,6,12,\ldots$,

The exponent is $n-1$ because the first term contains zero multiplications by $r$.

Worked example: find a distant term

Finding a common ratio from two terms

Suppose $t_3=18$ and $t_6=486$. Since three ratio steps separate the terms,

Thus $r^3=27$ and $r=3$. Then $18=ar^2=9a$, so $a=2$. The explicit rule is

Counting the number of steps between positions prevents an off-by-one exponent.

Behaviour for different ratios

• $r>1$: positive terms grow in magnitude.
• $0<r<1$: positive terms decay toward zero.
• $r=1$: the sequence is constant.
• $r<0$: signs alternate.
• $r=-1$: values alternate between two opposites.
• $|r|>1$: magnitude grows; $|r|<1$: magnitude shrinks.

The sign and magnitude of $r$ describe different features.

Connection to exponential functions

The points $(n,t_n)$ lie on an exponential curve, but a sequence normally uses positive integer inputs only. Geometric sequences are discrete exponential relations, just as arithmetic sequences are discrete linear relations.

Constant ratios in a table suggest an exponential model; constant differences suggest a linear model.

Modelling applications

Geometric sequences model repeated percentage change, bouncing heights, multi-stage dilution, population snapshots, and compound growth at fixed intervals.

If a quantity retains $85\%$ each period, then $r=0.85$. If it increases by $85\%$, then $r=1.85$. The words retains and increases by lead to different factors.

Solving for a term number

To find when $5(2)^{n-1}=640$, divide by $5$:

Therefore $n-1=7$ and $n=8$. If the bases do not match conveniently, logarithms or a graph can solve for the exponent, followed by a domain check.

Common mistakes

Subtracting terms to test the pattern. Geometric sequences use ratios.

Using exponent $n$. Starting from $a$ requires $n-1$ ratio steps.

Confusing a percent change with a factor. A $12\%$ increase uses $1.12$; retaining $12\%$ uses $0.12$.

Ignoring a negative ratio. It causes alternating signs.

Accepting a fractional position. Sequence indices must follow the stated discrete domain.

Quick self-check

• Are consecutive ratios constant?
• What are the sign and magnitude of $r$?
• Did I use $n-1$ in the explicit rule?
• Does the rule reproduce several terms?
• Does the growth or decay match the context?
• Is a solved term number a valid integer?

Related topics

• Arithmetic Sequences
• Series
• Exponential Functions
• Compound Interest