13.3 Geometric sequences (Page 2/6)

Page 2 / 6

\begin{array}{l} a_{1} = - 2 \\ a_{2} = (- 2 \cdot 4) = - 8 \\ a_{3} = (- 8 \cdot 4) = - 32 \\ a_{4} = (- 32 \cdot 4) - 128 \end{array}

The first four terms are ${–2, –8, –32, –128} .$

How To

Given the first term and the common factor, find the first four terms of a geometric sequence.

Multiply the initial term, $a_{1},$ by the common ratio to find the next term, $a_{2} .$
Repeat the process, using $a_{n} = a_{2}$ to find $a_{3}$ and then $a_{3}$ to find $a_{4,}$ until all four terms have been identified.
Write the terms separated by commons within brackets.

Writing the terms of a geometric sequence

List the first four terms of the geometric sequence with $a_{1} = 5$ and $r = –2.$

Multiply $a_{1}$ by $- 2$ to find $a_{2} .$ Repeat the process, using $a_{2}$ to find $a_{3},$ and so on.

\begin{array}{l} a_{1} = 5 \\ a_{2} = - 2 a_{1} = - 10 \\ a_{3} = - 2 a_{2} = 20 \\ a_{4} = - 2 a_{3} = - 40 \end{array}

The first four terms are ${5, –10, 20, –40} .$

Got questions? Get instant answers now!

Try It

List the first five terms of the geometric sequence with $a_{1} = 18$ and $r = \frac{1}{3} .$

${18, 6, 2, \frac{2}{3}, \frac{2}{9}}$

Got questions? Get instant answers now!

Using recursive formulas for geometric sequences

A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.

A General Note

Recursive formula for a geometric sequence

The recursive formula for a geometric sequence with common ratio $r$ and first term $a_{1}$ is

a_{n} = r a_{n - 1}, n \geq 2

How To

Given the first several terms of a geometric sequence, write its recursive formula.

State the initial term.
Find the common ratio by dividing any term by the preceding term.
Substitute the common ratio into the recursive formula for a geometric sequence.

Using recursive formulas for geometric sequences

Write a recursive formula for the following geometric sequence.

{6, 9, 13.5, 20.25, ...}

The first term is given as 6. The common ratio can be found by dividing the second term by the first term.

r = \frac{9}{6} = 1.5

Substitute the common ratio into the recursive formula for geometric sequences and define $a_{1} .$

\begin{array}{l} a_{n} = r a_{n - 1} \\ a_{n} = 1.5 a_{n - 1} for n \geq 2 \\ a_{1} = 6 \end{array}

Got questions? Get instant answers now!

A General Note

Do we have to divide the second term by the first term to find the common ratio?

No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.

Try It

Write a recursive formula for the following geometric sequence.

{2, \frac{4}{3}, \frac{8}{9}, \frac{16}{27}, ...}

$\begin{array}{l} a_{1} = 2 \\ a_{n} = \frac{2}{3} a_{n - 1} for n \geq 2 \end{array}$

Got questions? Get instant answers now!

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.

a_{n} = a_{1} r^{n - 1}

Let’s take a look at the sequence ${18, 36, 72, 144, 288, ...} .$ 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

a_{n} = 18 \cdot 2^{n - 1}

The graph of the sequence is shown in [link] .

A General Note

Explicit formula for a geometric sequence

The n th term of a geometric sequence is given by the explicit formula :

a_{n} = a_{1} r^{n - 1}

Writing terms of geometric sequences using the explicit formula

Given a geometric sequence with $a_{1} = 3$ and $a_{4} = 24,$ find $a_{2} .$

The sequence can be written in terms of the initial term and the common ratio $r .$

3, 3 r, 3 r^{2}, 3 r^{3}, ...

Find the common ratio using the given fourth term.

\begin{array}{l} a_{n} = a_{1} r^{n - 1} \\ a_{4} = 3 r^{3} & Write the fourth term of sequence in terms of α_{1} and r \\ 24 = 3 r^{3} & Substitute 24 for a_{4} \\ 8 = r^{3} & Divide \\ r = 2 & Solve for the common ratio \end{array}

Find the second term by multiplying the first term by the common ratio.

\begin{array}{l} a_{2} & = 2 a_{1} \\ = 2 (3) \\ = 6 \end{array}

Got questions? Get instant answers now!

Questions & Answers

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

what is labour ?

Lambiv

how will I do?

Venny Reply

how is the graph works?I don't fully understand

Rezat Reply

information

Eliyee

devaluation

Eliyee

WARKISA

hi guys good evening to all

Lambiv

multiple choice question

Aster Reply

appreciation

Eliyee

explain perfect market

Lindiwe Reply

In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

What is ceteris paribus?

Shukri Reply

other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

Kelo

yes,thank you

Shukri

Can I ask you other question?

Shukri

what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

Ezea

Shukri

how do you save a country economic situation when it's falling apart

Lilia Reply

what is the difference between economic growth and development

Fiker Reply

Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

Shukri

production function means

Jabir

What do you think is more important to focus on when considering inequality ?

Abdisa Reply

any question about economics?

Awais Reply

sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

Awais

thank you so much 👍 sir

Asui

In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

Feyisa Reply

Answer

Feyisa

Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

Gsbwnw Reply

suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

Abdureman

types of unemployment

Yomi Reply

What is the difference between perfect competition and monopolistic competition?

Mohammed

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 2

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask

	4 Using explicit formulas for geometric sequences By OpenStax Read Online Course
	2 Writing terms of geometric sequences By OpenStax Read Online Course
	19 Biology 19 The Evolution of Populations MCQ By OpenStax Start Quiz
	4 Microeconomics 04 Labor Financial Markets By OpenStax Start Flashcards
	Biology 302 Final By Dravida Mahadeo-J... Start Quiz
	Business fundamentals By OpenStax Read Online Course
	7 Biology 07 Cellular Respiration MCQ By OpenStax Start Quiz
©flickr:	Spanish Test By Tess Armstrong Start Quiz
	Kira Kira Test By Briana Knowlton Start Quiz
	1 Neuroscience Exam 2004 1 By David Corey Start Exam
	Biology Final By Anonymous User Start Quiz
	Precalculus By OpenStax Read Online Course