13.3 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

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

Desalegn Reply

U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

Jan Reply

this is the study of how the society manages it's scarce resources

Belonwu

what is macroeconomic

John Reply

macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

husaini Reply

what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

Petrus Reply

what is economic.

Milan Reply

scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

Zarshad Reply

Why qualify 28 supplies

Milan

what are explicit costs

Nomsa Reply

out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

David Reply

economic overview notes

Amahle Reply

identify a demand and a supply curve

Salome Reply

i don't know

Parul

there's a difference

Aryan

Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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

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