Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section we will review sequences that grow in this way.

Terms of Geometric Sequences

Finding Common Ratios

The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.

Writing Terms of Geometric Sequences

Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[/latex] and the common ratio is [latex]r=4[/latex], we can find subsequent terms by multiplying [latex]-2\cdot 4[/latex] to get [latex]-8[/latex] then multiplying the result [latex]-8\cdot 4[/latex] to get [latex]-32[/latex] and so on.

[latex]\begin{align}&{a}_{1}=-2 \\ &{a}_{2}=\left(-2\cdot 4\right)=-8 \\ &{a}_{3}=\left(-8\cdot 4\right)=-32 \\ &{a}_{4}=\left(-32\cdot 4\right)=-128 \end{align}[/latex]

The first four terms are [latex]\left\{-2,-8,-32,-128\right\}[/latex].

Explicit Formulas for Geometric Sequences

Using Explicit Formulas for Geometric Sequences

Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.

[latex]{a}_{n}={a}_{1}{r}^{n - 1}[/latex]

Let’s take a look at the sequence [latex]\left\{18\text{, }36\text{, }72\text{, }144\text{, }288\text{, }…\right\}[/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is

[latex]{a}_{n}=18\cdot {2}^{n - 1}[/latex]

In real-world scenarios involving arithmetic sequences, we may need to use an initial term of [latex]{a}_{0}[/latex] instead of [latex]{a}_{1}[/latex]. In these problems we can alter the explicit formula slightly by using the following formula:

[latex]{a}_{n}={a}_{0}{r}^{n}[/latex]

The following video provides a short lesson on some of the topics covered in this lesson.

 

A couple decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund pays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in 6 years? In this section we will learn how to answer this question. To do so we need to consider the amount of money invested and the amount of interest earned.

Geometric Series

Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series. Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio, [latex]r[/latex]. We can write the sum of the first [latex]n[/latex] terms of a geometric series as

[latex]{S}_{n}={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+…+{a}_{1}{r}^{n - 1}[/latex].

Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first [latex]n[/latex] terms of a geometric series. We will begin by multiplying both sides of the equation by [latex]r[/latex].

[latex]r{S}_{n}={a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+…+{a}_{1}{r}^{n}[/latex]

Next, we subtract this equation from the original equation.

[latex]\begin{align}{S}_{n}&={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+…+{a}_{1}{r}^{n - 1} \\ -r{S}_{n}&=-\left({a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+…+{a}_{1}{r}^{n}\right) \\ \hline \left(1-r\right){S}_{n}&={a}_{1}-{a}_{1}{r}^{n}\end{align}[/latex]

Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for [latex]{S}_{n}[/latex], factor [latex]a_1[/latex] on the right hand side and divide both sides by [latex]\left(1-r\right)[/latex].

[latex]{S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\text{ r}\ne \text{1}[/latex]

Using the Formula for the Sum of an Infinite Geometric Series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first n terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is [latex]2+4+6+8+\dots[/latex].

This series can also be written in summation notation as [latex] \sum\limits _{k=1}^{\infty} 2k[/latex], where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.

Determining Whether the Sum of an Infinite Geometric Series is Defined

If the terms of an infinite geometric series approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:

[latex]1+0.2+0.04+0.008+0.0016+\dots[/latex]

The common ratio is [latex]r=0.2[/latex]. As n gets large, the values of of [latex]r^n[/latex] get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with [latex]-1<r<1[/latex] approach 0; the sum of a geometric series is defined when [latex]-1<r<1[/latex].

Finding Sums of Infinite Series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first n terms of a geometric series.

[latex]{S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}[/latex]

We will examine an infinite series with [latex]r=\frac{1}{2}[/latex]. What happens to [latex]r^n[/latex] as n increases?

[latex]\begin{align} &{\left(\frac{1}{2}\right)}^{2} = \frac{1}{4} \\&{\left(\frac{1}{2}\right)}^{3} = \frac{1}{8} \\&{\left(\frac{1}{2}\right)}^{4} = \frac{1}{16} \end{align}[/latex]

The value of [latex]r^n[/latex] decreases rapidly. What happens for greater values of n?

[latex]\begin{align} &{\left(\frac{1}{2}\right)}^{10} = \frac{1}{1\text{,}024} \\&{\left(\frac{1}{2}\right)}^{20} = \frac{1}{1\text{,}048\text{,}576} \\&{\left(\frac{1}{2}\right)}^{30} = \frac{1}{1\text{,}073\text{,}741\text{,}824} \end{align}[/latex]

As n gets large, [latex]r^n[/latex] gets very small. We say that as n increases without bound, [latex]r^n[/latex] approaches 0. As [latex]r^n[/latex] approaches 0, [latex]1-r^n[/latex] approaches 1. When this happens the numerator approaches [latex]a_1[/latex]. This gives us the formula for the sum of an infinite geometric series.

 

Annuities

At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.

We can find the value of the annuity right after the last deposit by using a geometric series with [latex]{a}_{1}=50[/latex] and [latex]r=100.5\%=1.005[/latex]. After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.

We can find the value of the annuity after [latex]n[/latex] deposits using the formula for the sum of the first [latex]n[/latex] terms of a geometric series. In 6 years, there are 72 months, so [latex]n=72[/latex]. We can substitute [latex]{a}_{1}=50, r=1.005,[/latex] and [latex]n=72[/latex] into the formula, and simplify to find the value of the annuity after 6 years.

[latex]{S}_{72}=\dfrac{50\left(1-{1.005}^{72}\right)}{1 - 1.005}\approx 4\text{,}320.44[/latex]

After the last deposit, the couple will have a total of $4,320.44 in the account. Notice, the couple made 72 payments of $50 each for a total of [latex]72\left(50\right) = $3,600[/latex]. This means that because of the annuity, the couple earned $720.44 interest in their college fund.

Key Equations

recursive formula for [latex]nth[/latex] term of a geometric sequence	[latex]{a}_{n}=r{a}_{n - 1},n\ge 2[/latex]
explicit formula for [latex]nth[/latex] term of a geometric sequence	[latex]{a}_{n}={a}_{1}{r}^{n - 1}[/latex]
sum of the first [latex]n[/latex] terms of a geometric series	[latex]{S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r} , r\ne 1[/latex]
sum of an infinite geometric series with [latex]-1<r<1[/latex]	[latex]{S}_{n}=\dfrac{{a}_{1}}{1-r}[/latex]

Key Concepts

• A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
• The constant ratio between two consecutive terms is called the common ratio.
• The common ratio can be found by dividing any term in the sequence by the previous term.
• The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.
• A recursive formula for a geometric sequence with common ratio [latex]r[/latex] is given by [latex]{a}_{n}=r{a}_{n - 1}[/latex] for [latex]n\ge 2[/latex] .
• As with any recursive formula, the initial term of the sequence must be given.
• An explicit formula for a geometric sequence with common ratio [latex]r[/latex] is given by [latex]{a}_{n}={a}_{1}{r}^{n - 1}[/latex].
• In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}{r}^{n}[/latex].

• The sum of the terms in a sequence is called a series.
• A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum.
• The sum of the terms in a geometric sequence is called a geometric series.
• The sum of the first [latex]n[/latex] terms of a geometric series can be found using a formula.
• The sum of an infinite series exists if the series is geometric with [latex]-1<r<1[/latex].
• If the sum of an infinite series exists, it can be found using a formula.
• An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series.

Glossary

annuity an investment in which the purchaser makes a sequence of periodic, equal payments

common ratio the ratio between any two consecutive terms in a geometric sequence

diverge a series is said to diverge if the sum is not a real number

geometric sequence a sequence in which the ratio of a term to a previous term is a constant

geometric series the sum of the terms in a geometric sequence

infinite series the sum of the terms in an infinite sequence

series the sum of the terms in a sequence

summation notation a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series