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Geometric Sequences

Identify geometric sequences.

Learning Objectives

After completing this section, you should be able to:

Identify geometric sequences.
Find a given term in a geometric sequence.
Find the $n$th term of a geometric sequence.
Find the sum of a finite geometric sequence.
Use geometric sequences to solve real-world applications.

Identifying Geometric Sequences

We know what a sequence is, but what makes a sequence a geometric sequence? In an arithmetic sequence, each term is the previous term plus the constant difference. So, you add a (possibly negative) number at each step. In a geometric sequence, though, each term is the previous term multiplied by the same specified value, called the common ratio. In the sequence $\{3,6,12,24,48,96,192,384,728,1456\}$ the common ratio is 2. To see the difference between an arithmetic sequence and geometric sequence, examine these two sequences (Figures 3.52 and 3.53).

Each term in this arithmetic sequence is the previous term plus 5.

Each term in this geometric sequence is the previous term times 2.

In the sequence $\{3,6,12,24,48,96,192,384,728,1456\}$, the numbers get big fairly quickly, and stay positive. However, that’s not always the case with geometric sequences. Depending on the value of the common ratio, the terms could increase each time (like in the one shown in ), or the terms can get smaller each time, or the terms can alternate between positive and negative values. It all depends on the value of the common ratio, $r$.

Consider this geometric sequence: \[\{5,15,45,135,405,2025...\}\]

Each term is the previous term times 5, which means the common ratio is 5. This common ratio is larger than 1, and so the terms increase each time. Now, look at this geometric sequence: \[\{2,-6,18,-54,162,-486,1458...\}\]

Each term is the previous term times −3, and the sign of the terms alternate from positive to negative. Then, there’s this geometric sequence: \[\{9,3,1,\frac{1}{3},\frac{1}{9},\frac{1}{27}...\}\]

$i$, Place in Sequence	${a}_{i},{i}^{th}$,Term	Value of Term	Term Written as ${a}_{1}\times {r}^{i-1}$
1	${a}_{1}$	3	$3\times {2}^{0}$
2	${a}_{2}$	6	$3\times {2}^{1}$
3	${a}_{3}$	12	$3\times {2}^{2}$
4	${a}_{4}$	24	$3\times {2}^{3}$
5	${a}_{5}$	48	$3\times {2}^{4}$
$i$	${a}_{i}$		$3\times {2}^{i-1}$

Condensed — the full section is in OpenStax Contemporary Mathematics.

Finding the Sum of a Finite Geometric Sequence

As with arithmetic sequences, it is possible to add the terms of the geometric sequence. Like arithmetic sequences, the formula for the finite sum of the terms of a geometric sequence has a straightforward formula.

Calculating the Sum of a Finite Geometric Sequence

Try it.

What is the sum of the first 13 terms of the geometric sequence with first term ${a}_{1}=5$ and common ratio $r=3$?
What is the sum of the first 7 terms of the geometric sequence with first term ${a}_{1}=16$ and common ratio $r=\frac{1}{8}$?

Solution

Using ${a}_{1}=5$, $r=3$, and $n=13$, we find that the sum is: $\begin{array}{l}{S}_{13}={a}_{1}(\frac{1-{r}^{13}}{1-r})=5(\frac{1-{3}^{13}}{1-3})=5(\frac{1-1,594,323}{-2}) \\ =5(\frac{-1,594,322}{-2})=5(797,161)=3,985,805\end{array}$
The sum of the first 13 terms of this geometric sequence is 3,985,805.
Using ${a}_{1}=16$, $r=\frac{1}{8}$, and $n=7$, we find that the sum is: \[\begin{array}{l}{s}_{7}={a}_{1}(\frac{1-{r}^{7}}{1-r})=16(\frac{1-{(\frac{1}{8})}^{7}}{1-(\frac{1}{8})})=16(\frac{1-(\frac{1}{2,097,152})}{\frac{7}{8}}) \\ =16(\frac{\frac{2,097,151}{2,097,152}}{\frac{7}{8}})\ =16(\frac{299,593}{262,144})=\frac{299,593}{16,384}=18.2857\end{array}\]
The sum of the first 7 terms of this geometric sequence is $18.2857$.

Using Geometric Sequences to Solve Real-World Applications

Geometric sequences have a multitude of applications, one of which is compound interest. Compound interest is something that happens to money deposited into an account, be it savings or an individual retirement account, or IRA. The interest on the account is calculated and added to the account at regular intervals. This means the interest that was earned later gains its own interest. This allows the money to grow faster. If that interest is added every month, we say it is compounded monthly. If the interest is added daily, then we say it is compounded daily. The amount of money that is deposited into the account is called the principal and is denoted $P$. The account earns money on that principal. The amount it earns is a percentage of the money in the account. The interest rate, expressed as a decimal, is denoted $r$.

Calculating Interest Compounded Yearly

Try it.

Daryl deposits $1,000 in an account earning $4\%$ interest compounded yearly. How much money is in the account after 25 years?

Solution

Using $A=P{(1+r)}^{t}$ with $P=1000$, $r=0.04$, and $t=25$, we find that $A=P{(1+r)}^{t}=1,000\times {(1+0.04)}^{25}=1,000\times {(1.04)}^{25}=1,000\times 2.66583633=2,665.85$. After 25 years, there is $\$2,665.84$ in the account.

Another application of geometric sequences is exponential growth. This arises in biology quite frequently, especially in relation to bacterial cultures, but also with other organism population models. In bacterial cultures, the time it takes the population to double is often recorded. This time to double is the same, regardless of how big the population gets. So, if the population doubles after 3 hours, it doubles again after another 3 hours, and again after another 3 hours, and so on. Put into geometric sequence language, it has a common ratio of 2.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

A special form of a sequence is a geometric sequence. Each geometric sequence is determined by its first term and its constant ratio. Any term in a geometric sequence is determined by multiplying the constant ratio to the preceding term.
If the first term and the constant ratio of a geometric sequence are known, then any term of the sequence can be found directly.
Because geometric sequences follow such a strict pattern, the sum of the first $n$ terms of a geometric sequence can be determined with the formula ${s}_{n}={a}_{1}(\frac{1-{r}^{n-1}}{1-r})$.
Finding the sum of a finite geometric sequence
Applying arithmetic sequences

Encryption Throughout History

Encryption began at least as far back as the Roman Empire. During the reign of Caesar, a particular cypher was used, fittingly named the Caesar Cypher. This encryption process granted the Romans a great tactical advantage. Even if a message was intercepted, it would not make sense to the person intercepting the message.

Find four instances when encryption was used and cracked over the course of history.

Projects

The golden ratio has been used in art and architecture as far back as ancient Greece (possibly further). It also appears in South America (Incan architecture). Find five instances of the use of the golden ratio in art or architecture and describe its use in each of those instances.

Budgeting either is, or will shortly be, an important aspect of your life. Managing money well reduces stress in your life, and provides space for planning for future expenses, such as vacations or home improvements.

Imagine your life 10 years from now. Estimate your monthly income. Identify expenses you will encounter monthly (mortgage or rent, car payment, insurance, entertainment, etc.). Decide on an amount you plan to save monthly (this is treated as an expense). Create a spreadsheet with those values. Record your monthly net income (your income minus your expenses). Determine how much money you will have saved over the course of 5 years (ignore interest). Write a reflection on your anticipated financial health.

The value of pi is the ratio of the circumference of a circle to the diameter of the circle. It is also equal to the ratio of the area of the circle to the square of the radius of the square.

Research three ways to physically estimate pi.

Estimate pi using all three processes you found.

Present your process and solutions in class.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

Geometric sequence.
Finding an arbitrary term in a geometric sequence.
Constant ratio.
Finding the sum of a finite geometric sequence.
Applying arithmetic sequences.

Practice (6)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

For each sequence, determine if the sequence is a geometric sequence. If so, identify the common ratio.
1. $\{5,20,80,320,1,280,5,120,20,480,...\}$
2. $\{-3,6,-12,24,11,33\}$
3. $\{4,2,1,\frac{1}{2},\frac{1}{4},\frac{1}{8},...\}$
Откриј одговор.
1. In the sequence $\{5,20,80,320,1,280,5,120,20,480,...\}$, the jump from 5 to 20 is a multiplication by 4, as is the next jump to 80, and the next to 320. Each term is 4 times the previous term. Since each term is 4 times the previous, this is a geometric sequence. The common ratio is 4.
2. In the sequence $\{-3,6,-12,24,11,33\}$, notice that 6 is −3 times −2. The jump from 6 to −12 is another multiplication by negative. So, if this is a geometric sequence, each term should be the previous term times −2. But the change from 24 to 11 is not a multiplication by −2, This means the sequence is not a geometric sequence.
3. In the sequence $\{4,2,1,\frac{1}{2},\frac{1}{4},\frac{1}{8},...\}$, the change from 4 to 2 is a multiplication by $\frac{1}{2}$, as is the next jump, from 2 to 1, as is the next from 1 to $\frac{1}{2}$. Each term is $\frac{1}{2}$ times the previous term. Since each term is $\frac{1}{2}$ times the previous, this is a geometric sequence. The common ratio is $\frac{1}{2}$.
In the following geometric sequences, determine the indicated term of the geometric sequence with a given first term and common ratio.
1. Determine the $9\text{th}$ term of the geometric sequence with ${a}_{1}\times 6$ and $r=3$.
2. Determine the $11\text{th}$ term of the geometric sequence with ${a}_{1}=2$ and $r=-5$.
Откриј одговор.
1. Using ${a}_{n}={a}_{1}{r}^{n-1}$ with ${a}_{1}=6$, $r=3$, and $n=9$, we calculate \[{a}_{9}={a}_{1}{r}^{9-1}=6\times {(3)}^{9-1}=6\times {(3)}^{8}=6\times 6561=39366\].
  The $9\text{th}$ term of the geometric sequence with ${a}_{1}=6$ and $r=3$ is ${a}_{9}=39366$.
2. Using ${a}_{n}={a}_{1}{r}^{n-1}$ with ${a}_{1}=2$, $r=-5$, and $n=11$, we calculate \[{a}_{11}={a}_{1}{r}^{11-1}=2\times {(-5)}^{11-1}=2\times {(-5)}^{10}=2\times 9,765,625=19,531,250\].
1. What is the sum of the first 13 terms of the geometric sequence with first term ${a}_{1}=5$ and common ratio $r=3$?
2. What is the sum of the first 7 terms of the geometric sequence with first term ${a}_{1}=16$ and common ratio $r=\frac{1}{8}$?
Откриј одговор.
1. Using ${a}_{1}=5$, $r=3$, and $n=13$, we find that the sum is: $\begin{array}{l}{S}_{13}={a}_{1}(\frac{1-{r}^{13}}{1-r})=5(\frac{1-{3}^{13}}{1-3})=5(\frac{1-1,594,323}{-2}) \\ =5(\frac{-1,594,322}{-2})=5(797,161)=3,985,805\end{array}$
  The sum of the first 13 terms of this geometric sequence is 3,985,805.
2. Using ${a}_{1}=16$, $r=\frac{1}{8}$, and $n=7$, we find that the sum is: \[\begin{array}{l}{s}_{7}={a}_{1}(\frac{1-{r}^{7}}{1-r})=16(\frac{1-{(\frac{1}{8})}^{7}}{1-(\frac{1}{8})})=16(\frac{1-(\frac{1}{2,097,152})}{\frac{7}{8}}) \\ =16(\frac{\frac{2,097,151}{2,097,152}}{\frac{7}{8}})\ =16(\frac{299,593}{262,144})=\frac{299,593}{16,384}=18.2857\end{array}\]
  The sum of the first 7 terms of this geometric sequence is $18.2857$.
Daryl deposits $1,000 in an account earning $4\%$ interest compounded yearly. How much money is in the account after 25 years?

Откриј одговор.

Using $A=P{(1+r)}^{t}$ with $P=1000$, $r=0.04$, and $t=25$, we find that $A=P{(1+r)}^{t}=1,000\times {(1+0.04)}^{25}=1,000\times {(1.04)}^{25}=1,000\times 2.66583633=2,665.85$. After 25 years, there is $\$2,665.84$ in the account.
When Escherichia coli (E. coli) is in a broth culture at 37°C, the population of E. coli doubles in number with 30 organisms, how many E. coli bacteria are present in the culture after 16 hours?

Откриј одговор.

Since the population is doubling every 20 minutes, this is a geometric sequence situation with common ratio $r=2$. The culture begins with 30 organisms, so ${a}_{1}=30$. The time,16 hours, is 48 twenty-minute periods, so we’re looking for the 48th term in the sequence. Using these values in the geometric sequence formula gives
${a}_{48}={a}_{1}{r}^{n-1}=30\times {2}^{48-1}=30\times {2}^{47}=30\times (1.40737\times {10}^{14})=4.22212\times {10}^{15}$.
So, after 16 hours, the culture contains $4.22212\times {10}^{15}$ E. coli organisms. That’s more than 4,000 trillion bacteria.
A player places one grain of rice on the first square of a chess board. On the second square, the player places 2 grains of rice. On the third square, the player places 4 grains of rice. On each successive square of the board, the player doubles the number of grains of rice placed on the chess board. When the player places the last rice on the 64th square, how many total grains of rice have been placed on the board?

Откриј одговор.

Since the number of grains of rice is doubled at each step, this is a geometric sequence with first term ${a}_{1}=1$ and common ratio $r=2$. Rice is placed on 64 total squares, so we want the sum of the first 64 terms. Using this information and the formula, the total number of grains of rice on the board will be: $\begin{array}{l}{s}_{64}={a}_{1}(\frac{1-{r}^{n-1}}{1-r})=1\times (\frac{1-{2}^{64-1}}{1-2})=(\frac{1-{2}^{63}}{-1})=-(1-{2}^{63}) \\ =-(-9.2233720369\times {10}^{18})=9.2233720369\times {10}^{18}\end{array}$
That’s a 20-digit number!

Symbols used here

imaginary unit
i² = −1.

\neq

not equal
The two sides are different.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

\prod_{k=1}^{n} a_k

product
Multiply a_k for k = 1 up to n.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

a \mid b,\ \gcd(a,b)

divides, greatest common divisor
b is a multiple of a; the largest number dividing both.

\varphi(n),\ \pi(x)

Euler's totient, prime-counting function
Count of 1..n coprime to n; number of primes up to x.

\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n

integers modulo n
The remainders 0…n−1 with clock arithmetic.

a \bmod n

remainder
What is left after dividing a by n.

How to: Geometric Sequences

Identify geometric sequences.
Find a given term in a geometric sequence.
Find the
Find the sum of a finite geometric sequence.
Use geometric sequences to solve real-world applications.
In the sequence
In the sequence
In the sequence

Questions people ask

Why are primes so important?

Every integer factors into primes in exactly one way, so primes are the atoms of multiplication. Cryptography relies on that factoring being easy to state and hard to do.

How do I tell whether a big number is prime?

Trial division up to the square root works for small numbers. For large ones, probabilistic tests (Miller–Rabin) give an answer that is wrong with negligible probability, and deterministic tests (AKS) exist but are slower.

Покушај сам.

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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\(i\), Place in Sequence	\({a}_{i},{i}^{th}\),Term	Value of Term	Term Written as \({a}_{1}\times {r}^{i-1}\)
1	\({a}_{1}\)	3	\(3\times {2}^{0}\)
2	\({a}_{2}\)	6	\(3\times {2}^{1}\)
3	\({a}_{3}\)	12	\(3\times {2}^{2}\)
4	\({a}_{4}\)	24	\(3\times {2}^{3}\)
5	\({a}_{5}\)	48	\(3\times {2}^{4}\)
\(i\)	\({a}_{i}\)		\(3\times {2}^{i-1}\)