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

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

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

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

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}

13. Sequences, Series, and the Binomial Theorem

Geometric Sequences

13. 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. To find r, divide any 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.

To identify the common ratio in a geometric sequence, divide any term by the previous term. Mathematically, this is expressed as:

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

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, subsequent terms can be found 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 the common ratio is:

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

Using this, the next terms are calculated as:

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

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

Understanding geometric sequences is essential for analyzing patterns where multiplication or division defines the progression of terms. Recognizing the common ratio allows for predicting future terms and solving problems involving exponential growth or decay.

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.
$eighteen fifths comma six fifths comma two fifths comma 2 over 15 comma dot dot dot$

$3$

$\frac{1}{3}$

$\frac{1}{5}$

$- \frac{12}{5}$

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

Find the common ratio for the geometric sequence.
$6, - 3 \sqrt{2}, 3, - \frac{3 \sqrt{2}}{2}, \dots$

$- \sqrt{2}$

$- \frac{\sqrt{2}}{4}$

$- 2 \sqrt{2}$

$- \frac{\sqrt{2}}{2}$

Problem

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

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

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

$- 8, \frac{1}{4}, - \frac{1}{2}, 1, \dots$

$- \frac{1}{2}, 4, - 32, 256, \dots$

Problem

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

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

$\frac{3}{4}, \frac{1}{2}, \frac{1}{3}, \frac{2}{9}, \dots$

$three fourths comma 2 comma eight thirds comma 32 over 9 comma dot dot dot$

$two thirds comma one half comma three eighths comma 9 over 32 comma dot dot dot$

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 one less than the term’s position. 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 the appropriate power: the second term is 3 × 2¹, the third term is 3 × 2², and the fourth term is 3 × 2³. This pattern confirms the formula, where the exponent is always one less than the term number.

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

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

In another example, for the sequence 8, 24, 72, ..., the first term is 8 and the common ratio is 3, so the nth term formula is:

\[a_n = 8 \times 3^{n-1}\]

For sequences where the common ratio is a fraction, such as 16/27, 8/9, 3/4, 2, the common ratio is found by dividing a term by its previous term. For example, dividing 8/9 by 16/27 involves multiplying 8/9 by the reciprocal of 16/27, resulting in:

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

Thus, the nth term formula for this sequence is:

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

Understanding the general term formula for geometric sequences allows for quick computation of any term without listing all previous terms. This formula is essential for solving problems involving exponential growth or decay, financial calculations, and various applications in science and engineering.

Problem

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

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

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

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

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

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

In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio, denoted as r. Given that the third term (a₃) is 6 and the sixth term (a₆) is ¾, we can determine the common ratio and subsequently find the fifth term (a₅).

Since the terms are not consecutive, we recognize that moving from the third term to the sixth term involves multiplying by the common ratio three times. This relationship can be expressed algebraically as:

\[a_6 = a_3 \times r^3\]

Substituting the known values:

\[\frac{3}{4} = 6 \times r^3\]

To isolate r³, divide both sides by 6:

\[r^3 = \frac{3}{4} \times \frac{1}{6} = \frac{3}{24} = \frac{1}{8}\]

Taking the cube root of both sides gives the common ratio:

\[r = \sqrt[3]{\frac{1}{8}} = \frac{1}{2}\]

With the common ratio determined as ½, we can find the fifth term by multiplying the fourth term by r. First, calculate the fourth term:

\[a_4 = a_3 \times r = 6 \times \frac{1}{2} = 3\]

Then, the fifth term is:

\[a_5 = a_4 \times r = 3 \times \frac{1}{2} = \frac{3}{2}\]

Thus, the fifth term of the geometric sequence is $\frac{3}{2}$. This process highlights the importance of understanding how to manipulate exponents and roots to solve for unknowns in geometric sequences, especially when given non-consecutive terms.

example

General Term of a Geometric Sequence Example 2

Video duration:

General Term of a Geometric Sequence Example 2 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, expressed as a decimal, is the common ratio $r = 0.8$ of the geometric sequence.

The amount of medicine remaining after each hour can be calculated by multiplying the previous amount by the common ratio. For example, after one hour, the amount is $80 \times 0.8 = 64$ milligrams. After two hours, it becomes $64 \times 0.8 = 51.2$ milligrams, and this pattern continues. The general formula for the amount of medicine remaining after $n$ hours is given by the geometric sequence formula:

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

Applying this formula for five hours, we calculate:

\[a_5 = 80 \times 0.8^{4} = 80 \times 0.4096 = 32.768\]

However, since the problem tracks the amount after each hour starting from the initial dose, the amount after five hours is actually:

\[a_6 = 80 \times 0.8^{5} = 80 \times 0.32768 = 26.2144\]

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, the difference is 2, added each time. The key difference is multiplication for geometric sequences versus addition for arithmetic sequences.

To find the common ratio $r$ in a geometric sequence, you divide any term by the previous term. Mathematically, if $a_{n}$ is the $n$ th term and $a_{n-1}$ is the previous term, then the common ratio is $r = \frac{a_{n}}{a_{n-1}}$ . For example, in the sequence 3, 6, 12, 24, dividing 6 by 3 gives 2, so $r = 2$ . This ratio remains constant throughout the sequence.

The formula for the $n$ th term of a geometric sequence is $a_{n} = a_{1} r^{n-1}$ , where $a_{1}$ is the first term and $r$ is the common ratio. This formula allows you to find any term without calculating all 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 highlights the exponential nature of geometric sequences.

To calculate the 20th term of a geometric sequence efficiently, use the general term formula $a_{n} = a_{1} r^{n-1}$ . For example, if the first term $a_{1} = 3$ and the common ratio $r = 2$ , then the 20th term is $3 imes 2^{19}$ . This avoids multiplying repeatedly by 2 nineteen times and uses exponentiation instead, which is computationally efficient.

When the common ratio $r$ is a fraction, the general term formula remains the same: $a_{n} = a_{1} r^{n-1}$ . For example, if the first term is $$\frac{16}{27}$$ and the common ratio is $$\frac{3}{2}$$ , then the nth term is $$\frac{16}{27}$ $\times$ $\left$($\frac{3}{2}$$\right$)^{n-1}$ . You calculate powers of the fraction for the exponent and multiply by the first term to find any term in the sequence.

Exponents in geometric sequences represent how many times the common ratio is multiplied to reach a specific term. The nth term formula $a_{n} = a_{1} r^{n-1}$ shows that the common ratio $r$ is raised to the power of $n-1$ , indicating repeated multiplication. This exponential expression efficiently calculates terms far along in the sequence without repeated multiplication, emphasizing the connection between geometric sequences and exponential functions.

Your Intermediate Algebra tutors

Patrick Ford

Physics and Math Lead Instructor

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.
$0.8 comma 0.12 comma 0.018 comma 0.0027 comma dot dot dot$

Find the common ratio for the geometric sequence.
$6, - 3 \sqrt{2}, 3, - \frac{3 \sqrt{2}}{2}, \dots$

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

General Term of a Geometric Sequence Example 2

General Term of a Geometric Sequence Example 2 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 calculate the 20th term of a geometric sequence efficiently?

How do you write the general term formula for a geometric sequence with a fractional common ratio?

What role do exponents play in geometric sequences?

Your 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,…

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

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,…

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,…

Find the common ratio for the geometric sequence. 6,−32,3,−322,…6,-3\(\sqrt\)2,3,-\(\frac{3\sqrt2}{2}\),\(\ldots\)

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,…

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

General Term of a Geometric Sequence Example 2

General Term of a Geometric Sequence Example 2 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 calculate the 20th term of a geometric sequence efficiently?

How can you calculate the 20th term of a geometric sequence efficiently?

How do you write the general term formula for a geometric sequence with a fractional common ratio?

How do you write the general term formula for a geometric sequence with a fractional common ratio?

What role do exponents play in geometric sequences?

What role do exponents play in geometric sequences?

Your 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.
$0.8 comma 0.12 comma 0.018 comma 0.0027 comma dot dot dot$

Find the common ratio for the geometric sequence.
$6, - 3 \sqrt{2}, 3, - \frac{3 \sqrt{2}}{2}, \dots$

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 $a sub 5$ .