# 11.3 Geometric sequences  (Page 2/6)

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$\begin{array}{l}{a}_{1}=-2\hfill \\ {a}_{2}=\left(-2\cdot 4\right)=-8\hfill \\ {a}_{3}=\left(-8\cdot 4\right)=-32\hfill \\ {a}_{4}=\left(-32\cdot 4\right)-128\hfill \end{array}$

The first four terms are

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

1. Multiply the initial term, ${a}_{1},$ by the common ratio to find the next term, ${a}_{2}.$
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.
3. 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.

$\begin{array}{l}{a}_{1}=5\hfill \\ {a}_{2}=-2{a}_{1}=-10\hfill \\ {a}_{3}=-2{a}_{2}=20\hfill \\ {a}_{4}=-2{a}_{3}=-40\hfill \end{array}$

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

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

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

## 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.

## Recursive formula for a geometric sequence

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

${a}_{n}=r{a}_{n-1},n\ge 2$

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

1. State the initial term.
2. Find the common ratio by dividing any term by the preceding term.
3. 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.

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 ${a}_{1}.$

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.

Write a recursive formula for the following geometric sequence.

## 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 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

${a}_{n}=18·{2}^{n-1}$

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

## 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 $\text{\hspace{0.17em}}{a}_{1}=3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{a}_{4}=24,\text{\hspace{0.17em}}$ find ${a}_{2}.$

The sequence can be written in terms of the initial term and the common ratio $\text{\hspace{0.17em}}r.$

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

Find the common ratio using the given fourth term.

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

$\begin{array}{ll}{a}_{2}\hfill & =2{a}_{1}\hfill \\ \hfill & =2\left(3\right)\hfill \\ \hfill & =6\hfill \end{array}$

The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
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how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations