Geometric Sequences (College Board AP® Precalculus): Study Guide

Geometric sequences

What is a geometric sequence?

• A geometric sequence is a sequence in which successive terms have a common ratio

  • This means each term is obtained by multiplying the previous term by the same constant

  • This common ratio represents a constant proportional change

  • The common ratio is usually denoted by $r$

• E.g. the sequence $3,6,12,24,48,\dots$ is geometric with a common ratio of $r=2$

  • Each term is 2 times the previous term

    • $3\times 2=6$

    • $6\times 2=12$

    • $12\times 2=24$

    • etc.

• The common ratio can be any nonzero value:

  • $\mathit{r}\mathbf{>}\mathbf{1}$: the terms grow away from zero (for positive-valued sequences)

  • $\mathbf{0}\mathbf{<}\mathit{r}\mathbf{<}\mathbf{1}$: the terms shrink toward zero (for positive-valued sequences)

  • $\mathit{r}\mathbf{<}\mathbf{0}$: the terms alternate in sign

How can I find the common ratio of a geometric sequence?

• To find the common ratio of a geometric sequence

  • Divide any term by the term before it

    • $r=\frac{g_{n+1}}{g_n}$

• If the ratios between consecutive terms are not all equal, then the sequence is not geometric

What is the general term of a geometric sequence?

• The general term (also called the nth term) of a geometric sequence can be written in two forms

• Using the initial value

  • $g_n=g_0{r}^n$

    • where $g_0$ is the initial value (the term when $n=0$)

    • and $r$ is the common ratio

• Using any known term

  • $g_n=g_k{r}^{(n-k)}$

    • where $g_k$ is the value of the $\mathit{k}$th term

    • and $r$ is the common ratio

  • This form is useful when you don't know $g_0$ but you do know a different term

• Both forms express the same idea

  • Start from a known term and multiply by the common ratio the appropriate number of times

How do these formulas work in practice?

• E.g. a geometric sequence has initial value $g_0=5$ and a common ratio of $r=3$

  • The general term is $g_n=5·3^n$

  • So $g_0=5$, $g_1=5\times 3^1=15$, $g_2=5\times 3^2=45$, $g_3=5\times 3^3=135$, $\dots$

• Or e.g. you are told that $g_2=48$ and $r=\frac{1}{2}$

  • Using the second form: $g_n=48·{\left(\frac{1}{2}\right)}^{(n-2)}$

  • You can verify that to check the answer

    • $g_2=48·{\left(\frac{1}{2}\right)}^{2-2}=48·{\left(\frac{1}{2}\right)}^0=48·1=48$ ✓

  • And also find other terms

    • E.g. the initial term is $g_0=48·{\left(\frac{1}{2}\right)}^{0-2}=48·{\left(\frac{1}{2}\right)}^{-2}=48·4=192$

How does a geometric sequence grow compared to an arithmetic sequence?

• An increasing arithmetic sequence increases equally with each step

  • the same amount is added each time

• An increasing geometric sequence (with positive values) increases by a larger amount with each successive step

  • because the same ratio is applied to an ever-larger value

• E.g. consider the geometric sequence $2,6,18,54,162,\dots$ (ratio $r=3$)

  • The increases between terms are $4,12,36,108,\dots$

    • getting larger each time

• Compare this to the arithmetic sequence $2,6,10,14,18,\dots$ (difference $d=4$)

  • The increases between terms are always $4$

• This distinction between additive change (arithmetic) versus multiplicative change (geometric) is a key idea that carries over into the study of linear and exponential functions

What does the graph of a geometric sequence look like?

• Like all sequences, the graph of a geometric sequence consists of discrete points at whole number values of $n$

• For a geometric sequence with $\mathit{r}\mathbf{>}\mathbf{1}$ (and positive initial value), the points curve upward with increasing steepness

• For a geometric sequence with $\mathbf{0}\mathbf{<}\mathit{r}\mathbf{<}\mathbf{1}$ (and positive initial value), the points curve downward, getting closer and closer to zero