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Honors: 7. Sequences
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United States of AmericaFL
Grade 912

Honors: 7.03 Exponential functions as geometric sequences

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Lesson

Concept summary

In addition to the arithmetic sequence, there is another special type of sequence called a geometric sequence. These sequences have a constant multiplicative pattern.

Geometric sequence

A sequence of numbers in which each consecutive pair of numbers has a common ratio.

Example:

1, 3, 9, 27, 81,\ldots

Common ratio

The ratio of consecutive terms in a geometric sequence: r=\dfrac{a_n}{a_{n-1}}

Example:

1, 3, 9, 27, 81,\ldots has r=3

The nth term, a_n, of a geometric sequence is given by the explicit rule or general formula:

\displaystyle a_n=a_1 r^{n-1}
\bm{a_1}
The first term
\bm{r}
The common ratio

Note: other notations may be used such as T_n, u_n, b_n, etc.

Worked examples

Example 1

Identify whether each sequence is geometric. If it is a geometric sequence, write the explicit rule.

a

3.1,\,5.6,\,8.1,\,10.6,\, \ldots

Approach

The ratio of consecutive terms in a geometric sequence is always the same. That is, if the sequence a_1,\,a_2,\,a_3,\,a_4,\,\ldots is geometric, then \dfrac{a_2}{a_1}=\dfrac{a_3}{a_2}=\dfrac{a_4}{a_3}=\ldots

Solution

From the given sequence: a_1=3.1,\,a_2=5.6,\,a_3=8.1,\,a_4=10.6,\, \ldots

Observe that \dfrac{a_2}{a_1}=\dfrac{5.6}{3.1}=1.81 and \dfrac{a_3}{a_2}=\dfrac{8.1}{5.6}=1.45. So we have \dfrac{a_2}{a_1} \neq\dfrac{a_3}{a_2}. Because the ratio of consecutive terms for this sequence is not always the same this sequence is not geometric.

Reflection

Notice that a_2-a_1=5.6-3.1=2.5,\,a_3-a_2=8.1-5.6=2.5,\,a_4-a_3=10.6-8.1=2.5. The sequence has a common difference of 2.5. This tells us that the sequence is actually an arithmetic sequence.

b

6,\,18,\, 54,\,162,\, \ldots

Approach

If the sequence a_1,\,a_2,\,a_3,\,a_4,\,\ldots is geometric, then \dfrac{a_2}{a_1}=\dfrac{a_3}{a_2}=\dfrac{a_4}{a_3}=\ldots

The explicit rule for the geometric sequence is given by a_n = a_1 r^{n - 1} where a_1 is the first term in the sequence and r is the common ratio.

Solution

From the given sequence: a_1=6,\,a_2=18,\,a_3=54,\,a_4=162,\, \ldots

Check each ratio: \dfrac{a_2}{a_1}=\dfrac{18}{6}=3,\quad\dfrac{a_3}{a_2}=\dfrac{54}{18}=3,\quad\dfrac{a_4}{a_3}=\dfrac{162}{54}=3.

This sequence is geometric since each ratio is equal.

To write the explicit rule we can substitute a_1=6 and r=3 into a_n = a_1 r^{n - 1} which gives us a_n = 6(3)^{n - 1}

Reflection

Use parentheses around the common ratio to separate it from the first term: 6(3)^{n-1} vs. 63^{n-1}

c

-4,\,20,\, -100,\,500,\, \ldots

Approach

If the sequence a_1,\,a_2,\,a_3,\,a_4,\,\ldots is geometric, then \dfrac{a_2}{a_1}=\dfrac{a_3}{a_2}=\dfrac{a_4}{a_3}=\ldots

The explicit rule for the geometric sequence is given by a_n = a_1 r^{n - 1} where a_1 is the first term in the sequence and r is the common ratio.

Solution

From the given sequence: a_1=-4,\,a_2=20,\,a_3=-100,\,a_4=500,\, \ldots

Check each ratio: \dfrac{a_2}{a_1}=\dfrac{20}{-4}=-5,\quad\dfrac{a_3}{a_2}=\dfrac{-100}{20}=-5,\quad\dfrac{a_4}{a_3}=\dfrac{500}{-100}=-5

This sequence is geometric since each ratio is equal.

To write the explicit rule we can substitute a_1=-4 and r=-5 into a_n = a_1 r^{n - 1} which gives us a_n = -4(-5)^{n - 1}

Reflection

The explicit rule for any sequence with alternating signs will include a factor of (-1)^n somewhere.

Example 2

Consider the geometric sequence defined by: a_n=64 \left(\frac{1}{2} \right)^{n-1}

a

Identify the first term and common ratio.

Approach

We can use the explicit rule for a geometric sequence a_n=a_1r^{n-1} to identify the first term and common ratio.

Solution

a_1=64 and r=\dfrac{1}{2}.

Reflection

The common ratio will always be the number that is being raised to a power. The first term will always be the number that is not being raised to a power.

If it looks like there is no first term, then a_1=1, like in the sequence: 2^{n-1}

b

Plot the first four terms on a coordinate plane.

Approach

Each term can be written as a coordinate point in the form \left(n,a_n\right). To find the first four terms we can organize our work in an input-output table.

Solution

The first four terms of the geometric sequence are:

na_n
164
232
316
48

The graph of the first four terms of the geometric sequence is:

1
2
3
4
n
8
16
24
32
40
48
56
64
72
a_n

Reflection

The graph of a sequence should always be a represented with disconnected, or discrete, points since the term number is always a whole number.

c

Find a_{10}.

Approach

We can use our explicit rule to find the tenth term, a_{10}.

Solution

\displaystyle a_n\displaystyle =\displaystyle 64\left(\frac{1}{2}\right)^{n-1}Explicit rule
\displaystyle a_{10}\displaystyle =\displaystyle 64\left(\frac{1}{2}\right)^{(10-1)}Substitute 10 for n
\displaystyle a_{10}\displaystyle =\displaystyle 64\left(\frac{1}{2}\right)^{9}Evaluate the difference
\displaystyle a_{10}\displaystyle =\displaystyle \dfrac{1}{8}Evaluate the expression

Reflection

If we wanted, we could also continue the pattern in our table from part (b) to find the tenth term:

n12345678910
a_n6432168421\dfrac{1}{2}\dfrac{1}{4}\dfrac{1}{8}

Outcomes

MA.912.AR.5.4

Write an exponential function to represent a relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.

MA.912.AR.10.2

Given a mathematical or real-world context, write and solve problems involving geometric sequences.

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