9.3 Geometric sequences (Page 2/6)

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\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 ${5, –10, 20, –40} .$

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Try It

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

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

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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}

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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}, ...}

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

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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}

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Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

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