Find the common ratio for the geometric sequence. 18 5 , 6 5 , 2 5 , 2 15 , … \frac{18}{5},\frac65,\frac25,\frac{2}{15},\ldots

− 12 5 -\frac{12}{5} − 5 12

Write the first four terms of the following geometric sequence. a 1 = 3 4 , r = 2 3 a_1=\frac34,r=\frac23

3 4 , 1 2 , 1 3 , 2 9 , … \frac34,\frac12,\frac13,\frac29,\ldots 4 3 , 2 1 , 3 1 , 9 2 , …

3 4 , 1 2 , 3 8 , 9 32 , … \frac34,\frac12,\frac38,\frac{9}{32},\ldots

3 4 , 2 , 8 3 , 32 9 , … \frac34,2,\frac83,\frac{32}{9},\ldots 4 3 , 2 , 3 8 , 9 32 , …

2 3 , 1 2 , 3 8 , 9 32 , … \frac23,\frac12,\frac38,\frac{9}{32},\ldots 3 2 , 2 1 , 8 3 , 32 9 , …

Write a formula for the general or n t h ⁡ n^{\operatorname{\mathrm{th}}} term for each geometric sequence. 12 , − 6 , 3 , − 3 2 , … 12,-6,3,-\frac32,\ldots

a n = 12 ( − 1 2 ) n − 1 a_{n}=12\left(-\frac12\right)^{n-1}

a n = 12 ( − 2 ) n − 1 a_{n}=12\left(-2\right)^{n-1}

a n = 12 ( 1 2 ) n − 1 a_{n}=12\left(\frac12\right)^{n-1} a n = 12 ( 2 1 ) n − 1

a n = ( − 1 2 ) n − 1 a_{n}=\left(-\frac12\right)^{n-1} a n = ( − 2 1 ) n − 1

Write a formula for the general or n t h ⁡ n^{\operatorname{\mathrm{th}}} term for each geometric sequence. 0.8 , 0.32 , 0.128 , 0.0512 , … 0.8,0.32,0.128,0.0512,\ldots

a n = 0.8 ( 0.4 ) n − 1 a_{n}=0.8\left(0.4\right)^{n-1}

a n = 0.8 ( 2.5 ) n − 1 a_{n}=0.8\left(2.5\right)^{n-1}

a n = ( 0.4 ) n − 1 a_{n}=\left(0.4\right)^{n-1}

a n = ( 2.5 ) n − 1 a_{n}=\left(2.5\right)^{n-1}

15. Sequences, Series, and the Binomial Theorem

Geometric Sequences

15. Sequences, Series, and the Binomial Theorem

Geometric Sequences: Videos & Practice Problems

Video Lessons

Topic summary

Geometric sequences are defined by a constant multiplier called the common ratio (r) between consecutive terms. The general term formula is an = a1 rn-1, where a1 is the first term. Identifying r involves dividing a term by its previous term. This exponential expression highlights the power of the common ratio, enabling efficient calculation of any term without listing all preceding terms, essential for understanding sequences and series in algebra.

concept

Intro To Geometric Sequences

Video duration:

Intro To Geometric Sequences Video Summary

A geometric sequence is a type of number sequence where each term is found by multiplying the previous term by a constant value called the common ratio, denoted as r. This distinguishes geometric sequences from arithmetic sequences, where the pattern involves adding or subtracting a fixed number, known as the common difference.

For example, in an arithmetic sequence like 2, 4, 6, 8, the pattern is adding 2 each time. However, in a geometric sequence such as 2, 4, 8, 16, each term is multiplied by 2, making the common ratio r = 2. Similarly, sequences can also have common ratios less than 1, such as 9, 3, 1, 1/3, where each term is multiplied by 1/3 (or divided by 3).

To find the common ratio r in a geometric sequence, divide any term by the previous term:

\[r = \frac{a_n}{a_{n-1}}\]

where aₙ is the current term and aₙ₋₁ is the preceding term. This ratio remains constant throughout the sequence.

Once the common ratio is known, you can find subsequent terms by multiplying the last known term by r. For instance, if the first term a₁ is 5 and the second term a₂ is 20, then:

\[r = \frac{a_2}{a_1} = \frac{20}{5} = 4\]

Using this, the third term a₃ is:

\[a_3 = a_2 \times r = 20 \times 4 = 80\]

and the fourth term a₄ is:

\[a_4 = a_3 \times r = 80 \times 4 = 320\]

Understanding geometric sequences is essential for recognizing patterns where multiplication or division defines the progression of terms. This concept is widely applicable in fields such as finance, physics, and computer science, where exponential growth or decay occurs.

Problem

Identify if each sequence is arithmetic, geometric, or neither.
$7 comma 10 comma 16 comma 25 comma 37 comma dot dot dot$

Arithmetic

Geometric

Neither

Problem

Identify if each sequence is arithmetic, geometric, or neither.
$5 comma 11 comma 17 comma 23 comma 29 comma dot dot dot$

Arithmetic

Geometric

Neither

Problem

Identify if each sequence is arithmetic, geometric, or neither.
$10 comma 5 comma five halves comma five fourths comma dot dot dot$

Arithmetic

Geometric

Neither

Problem

Find the common ratio for the geometric sequence.
$3 comma 12 comma 48 comma 192 comma dot dot dot$

$3$

$4$

$6$

$9$

Problem

Find the common ratio for the geometric sequence.
$eighteen fifths comma six fifths comma two fifths comma 2 over 15 comma dot dot dot$

$3$

$$\frac$13$

$$\frac$15$

$-$\frac{12}{5}$$

Problem

Find the common ratio for the geometric sequence.
$0.8 comma 0.12 comma 0.018 comma 0.0027 comma dot dot dot$

$0.68$

$6.67$

$0.15$

$0.0034$

Problem

Write the first four terms of the following geometric sequence.
$a_{1} = - 8, r = - \frac{1}{2}$

$-8,4,-2,1,$\ldots$$

$-8,-$\frac{17}{2}$,-9,-$\frac{19}{2}$,$\ldots$$

$-8,$\frac$14,-$\frac$12,1,$\ldots$$

$-$\frac$12,4,-32,256,$\ldots$$

Problem

Write the first four terms of the following geometric sequence.
$a_{1} = \frac{3}{4}, r = \frac{2}{3}$

$three fourths comma one half comma three eighths comma 9 over 32 comma dot dot dot$

$\frac\)34,$\frac$12,$\frac$13,$\frac$29,\(\ldots$

$\frac\)34,2,$\frac$83,$\frac{32}{9}$,\(\ldots$

$\frac\)23,$\frac$12,$\frac$38,$\frac{9}{32}$,\(\ldots$

concept

General Term of a Geometric Sequence

Video duration:

General Term of a Geometric Sequence Video Summary

A geometric sequence is defined by a constant ratio, called the common ratio r, between consecutive terms. To find the general nth term of a geometric sequence, denoted as a_n, you use the first term a₁ and multiply it by the common ratio raised to the power of n - 1. This relationship can be expressed with the formula:

\[a_n = a_1 \times r^{n-1}\]

For example, consider the geometric sequence 3, 6, 12, 24. The first term a₁ is 3, and the common ratio r is found by dividing any term by its previous term, such as 6 ÷ 3 = 2. Each term can be rewritten as the first term multiplied by 2 raised to a power one less than the term’s position: the second term is 3 × 2¹, the third term is 3 × 2², and so on. This pattern confirms the formula for the nth term.

Using this formula, calculating large terms becomes efficient. For instance, the 20th term of the sequence is:

\[a_{20} = 3 \times 2^{19} = 1,572,864\]

When given a geometric sequence, you can determine the common ratio by dividing any term by its preceding term. For example, in the sequence ¹⁶/₂₇, ⁸/₉, ³/₄, 2, the common ratio is calculated as:

\[r = \frac{\frac{8}{9}}{\frac{16}{27}} = \frac{8}{9} \times \frac{27}{16} = \frac{3}{2}\]

Thus, the general nth term formula for this sequence is:

\[a_n = \frac{16}{27} \times \left(\frac{3}{2}\right)^{n-1}\]

Understanding and applying this formula allows for quick computation of any term in a geometric sequence without the need for repeated multiplication. This concept is fundamental in sequences and series, providing a powerful tool for analyzing exponential growth or decay patterns in various mathematical and real-world contexts.

Problem

Write a formula for the general or $n^{t h}$ term for each geometric sequence.
$12, - 6, 3, - \frac{3}{2}, \dots$

$a_{n} = 12 {(- 2)}^{n - 1}$

$a_{n} = 12 {(- \frac{1}{2})}^{n - 1}$

$a_{n}=12$\left$($\frac$12$\right$)^{n-1}$

$a_{n}=$\left$(-$\frac$12$\right$)^{n-1}$

Problem

Write a formula for the general or $n^{t h}$ term for each geometric sequence.
$0.8 comma 0.32 comma 0.128 comma 0.0512 comma dot dot dot$

$a_{n} = 0.8 {(2.5)}^{n - 1}$

$a_{n} = {(0.4)}^{n - 1}$

$a_{n} = {(2.5)}^{n - 1}$

$a_{n} = 0.8 {(0.4)}^{n - 1}$

Problem

Find the indicated term of the following geometric sequence.
$a_{n} = 10 (\frac{1}{5})^{n - 1}$
Find $a sub 4$ .

$2 over 125$

$2 over 25$

$8 over 25$

$2 over 625$

Problem

Find the indicated term of the following geometric sequence.
$a_{n} = 4 (0.3)^{n - 1}$
Find $a sub 5$ .

$0.0324$

$0.0097$

$0.0029$

$0.1620$

example

General Term of a Geometric Sequence Example 1

Video duration:

General Term of a Geometric Sequence Example 1 Video Summary

When a medicine decreases in the bloodstream by 20% each hour, the amount remaining follows a geometric sequence. Initially, the patient has 80 milligrams of the medicine in their system, which serves as the first term of the sequence, denoted as $a_1 = 80$. Since the medicine decreases by 20% every hour, 80% remains after each hour. This percentage can be expressed as the common ratio \(r = 0.8\( in decimal form.

The amount of medicine remaining after each hour is found by multiplying the previous amount by the common ratio. This relationship is described by the formula for the nth term of a geometric sequence:

\[a_n = a_1 \times r^{n-1}\]

Applying this formula, the amount of medicine after 5 hours is:

\[a_5 = 80 \times 0.8^{4}\]

Calculating the powers of 0.8 step-by-step, we find:

After 1 hour: \)80 \times 0.8 = 64\) mg
After 2 hours: $64 \times 0.8 = 51.2$ mg
After 3 hours: $51.2 \times 0.8 = 40.96$ mg
After 4 hours: $40.96 \times 0.8 = 32.768$ mg
After 5 hours: $32.768 \times 0.8 = 26.2144$ mg

Rounding to the nearest tenth, approximately 26.2 milligrams of the medicine remain in the bloodstream after five hours. This example illustrates how geometric sequences effectively model exponential decay processes, such as the reduction of medicine concentration in the body over time.

Here’s what students ask on this topic:

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio, denoted as $r$ . This contrasts with an arithmetic sequence, where each term is found by adding a constant difference to the previous term. For example, in the geometric sequence 2, 4, 8, 16, each term is multiplied by 2 to get the next term, so the common ratio $r =2$ . In an arithmetic sequence like 2, 4, 6, 8, each term increases by adding 2, which is the common difference. Understanding this difference helps in identifying the type of sequence and applying the correct formulas for finding terms.

To find the common ratio $r$ in a geometric sequence, you divide any term by the previous term. Mathematically, this is expressed as $r = \frac{an}{an-1}$ , where $an$ is the current term and $an-1$ is the previous term. For example, in the sequence 3, 6, 12, 24, dividing 6 by 3 gives 2, so the common ratio is 2. This ratio remains constant throughout the sequence, which is a defining property of geometric sequences.

The formula for the nth term $an$ of a geometric sequence is $an = a1 r^{n-1}$ , where $a1$ is the first term, $r$ is the common ratio, and $n$ is the term number. This formula allows you to find any term in the sequence without calculating all the previous terms. For example, if the first term is 3 and the common ratio is 2, the 4th term is $3 imes 2^{3} = 24$ . This formula is essential for efficiently working with geometric sequences.

To find a specific term in a geometric sequence using the nth term formula, you substitute the values of the first term $a1$ , the common ratio $r$ , and the term number $n$ into the formula $an = a1 r^{n-1}$ . For example, if the first term is 5, the common ratio is 4, and you want to find the 5th term, calculate $5 imes 4^{4} = 5 imes 256 = 1280$ . This method saves time compared to multiplying the common ratio repeatedly.

To determine if a sequence is geometric, check if the ratio between consecutive terms is constant. Calculate the ratio by dividing each term by the previous term. If all these ratios are equal, the sequence is geometric. For example, in the sequence 9, 3, 1, 1/3, dividing 3 by 9 gives 1/3, dividing 1 by 3 gives 1/3, and dividing 1/3 by 1 also gives 1/3. Since the ratio is consistently 1/3, this sequence is geometric with a common ratio of 1/3.

Your Beginning & Intermediate Algebra tutors

Geometric Sequences: Videos & Practice Problems

Intro To Geometric Sequences

Intro To Geometric Sequences Video Summary

Identify if each sequence is arithmetic, geometric, or neither.
$7 comma 10 comma 16 comma 25 comma 37 comma dot dot dot$

Identify if each sequence is arithmetic, geometric, or neither.
$5 comma 11 comma 17 comma 23 comma 29 comma dot dot dot$

Identify if each sequence is arithmetic, geometric, or neither.
$10 comma 5 comma five halves comma five fourths comma dot dot dot$

Find the common ratio for the geometric sequence.
$3 comma 12 comma 48 comma 192 comma dot dot dot$

Find the common ratio for the geometric sequence.
$eighteen fifths comma six fifths comma two fifths comma 2 over 15 comma dot dot dot$

Find the common ratio for the geometric sequence.
$0.8 comma 0.12 comma 0.018 comma 0.0027 comma dot dot dot$

Write the first four terms of the following geometric sequence.
$a_{1} = - 8, r = - \frac{1}{2}$

Write the first four terms of the following geometric sequence.
$a_{1} = \frac{3}{4}, r = \frac{2}{3}$

General Term of a Geometric Sequence

General Term of a Geometric Sequence Video Summary

Write a formula for the general or $n^{t h}$ term for each geometric sequence.
$12, - 6, 3, - \frac{3}{2}, \dots$

Write a formula for the general or $n^{t h}$ term for each geometric sequence.
$0.8 comma 0.32 comma 0.128 comma 0.0512 comma dot dot dot$

Find the indicated term of the following geometric sequence.
$a_{n} = 10 (\frac{1}{5})^{n - 1}$
Find $a sub 4$ .

Find the indicated term of the following geometric sequence.
$a_{n} = 4 (0.3)^{n - 1}$
Find $a sub 5$ .

General Term of a Geometric Sequence Example 1

General Term of a Geometric Sequence Example 1 Video Summary

Here’s what students ask on this topic:

What is a geometric sequence and how is it different from an arithmetic sequence?

How do you find the common ratio in a geometric sequence?

What is the formula for the nth term of a geometric sequence?

How can you use the nth term formula to find a specific term in a geometric sequence?

How do you determine if a sequence is geometric?

Your Beginning & Intermediate Algebra tutors

Geometric Sequences: Videos & Practice Problems

Intro To Geometric Sequences

Intro To Geometric Sequences Video Summary

Identify if each sequence is arithmetic, geometric, or neither.7,10,16,25,37,…7,10,16,25,37,\(\ldots\)

Identify if each sequence is arithmetic, geometric, or neither.5,11,17,23,29,…5,11,17,23,29,\(\ldots\)

Identify if each sequence is arithmetic, geometric, or neither.10,5,52,54,…10,5,\(\frac\)52,\(\frac\)54,\(\ldots\)

Find the common ratio for the geometric sequence.3,12,48,192,…3,12,48,192,\(\ldots\)

Find the common ratio for the geometric sequence.185,65,25,215,…\(\frac{18}{5}\),\(\frac\)65,\(\frac\)25,\(\frac{2}{15}\),\(\ldots\)

Find the common ratio for the geometric sequence.0.8,0.12,0.018,0.0027,…0.8,0.12,0.018,0.0027,\(\ldots\)

Write the first four terms of the following geometric sequence.a1=−8,r=−12a_1=-8,r=-\(\frac\)12

Write the first four terms of the following geometric sequence.a1=34,r=23a_1=\(\frac\)34,r=\(\frac\)23

General Term of a Geometric Sequence

General Term of a Geometric Sequence Video Summary

Write a formula for the general or nthn^{\(\operatorname{\mathrm{th}\)}} term for each geometric sequence.12,−6,3,−32,…12,-6,3,-\(\frac\)32,\(\ldots\)

Write a formula for the general or nthn^{\(\operatorname{\mathrm{th}\)}} term for each geometric sequence.0.8,0.32,0.128,0.0512,…0.8,0.32,0.128,0.0512,\(\ldots\)

Find the indicated term of the following geometric sequence.an=10(15)n−1a_{n}=10(\(\frac\)15)^{n-1} Find a4a_4.

Find the indicated term of the following geometric sequence.an=4(0.3)n−1a_{n}=4(0.3)^{n-1}Find a5a_5.

General Term of a Geometric Sequence Example 1

General Term of a Geometric Sequence Example 1 Video Summary

Here’s what students ask on this topic:

What is a geometric sequence and how is it different from an arithmetic sequence?

What is a geometric sequence and how is it different from an arithmetic sequence?

How do you find the common ratio in a geometric sequence?

How do you find the common ratio in a geometric sequence?

What is the formula for the nth term of a geometric sequence?

What is the formula for the nth term of a geometric sequence?

How can you use the nth term formula to find a specific term in a geometric sequence?

How can you use the nth term formula to find a specific term in a geometric sequence?

How do you determine if a sequence is geometric?

How do you determine if a sequence is geometric?

Your Beginning & Intermediate Algebra tutors

Identify if each sequence is arithmetic, geometric, or neither.
$7 comma 10 comma 16 comma 25 comma 37 comma dot dot dot$

Identify if each sequence is arithmetic, geometric, or neither.
$5 comma 11 comma 17 comma 23 comma 29 comma dot dot dot$

Identify if each sequence is arithmetic, geometric, or neither.
$10 comma 5 comma five halves comma five fourths comma dot dot dot$

Find the common ratio for the geometric sequence.
$3 comma 12 comma 48 comma 192 comma dot dot dot$

Find the common ratio for the geometric sequence.
$eighteen fifths comma six fifths comma two fifths comma 2 over 15 comma dot dot dot$

Find the common ratio for the geometric sequence.
$0.8 comma 0.12 comma 0.018 comma 0.0027 comma dot dot dot$

Write the first four terms of the following geometric sequence.
$a_{1} = - 8, r = - \frac{1}{2}$

Write the first four terms of the following geometric sequence.
$a_{1} = \frac{3}{4}, r = \frac{2}{3}$

Write a formula for the general or $n^{t h}$ term for each geometric sequence.
$12, - 6, 3, - \frac{3}{2}, \dots$

Write a formula for the general or $n^{t h}$ term for each geometric sequence.
$0.8 comma 0.32 comma 0.128 comma 0.0512 comma dot dot dot$

Find the indicated term of the following geometric sequence.
$a_{n} = 10 (\frac{1}{5})^{n - 1}$
Find $a sub 4$ .

Find the indicated term of the following geometric sequence.
$a_{n} = 4 (0.3)^{n - 1}$
Find $a sub 5$ .