# 13.3 Geometric sequences  (Page 2/6)

 Page 2 / 6
$\begin{array}{l}{a}_{1}=-2\hfill \\ {a}_{2}=\left(-2\cdot 4\right)=-8\hfill \\ {a}_{3}=\left(-8\cdot 4\right)=-32\hfill \\ {a}_{4}=\left(-32\cdot 4\right)-128\hfill \end{array}$

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.

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

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function