<< Chapter < Page Chapter >> Page >
a 1 = 2 a 2 = ( 2 4 ) = 8 a 3 = ( 8 4 ) = 32 a 4 = ( 32 4 ) 128

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

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.

a 1 = 5 a 2 = 2 a 1 = 10 a 3 = 2 a 2 = 20 a 4 = 2 a 3 = 40

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

Got questions? Get instant answers now!
Got questions? Get instant answers now!

List the first five terms of the geometric sequence with a 1 = 18 and r = 1 3 .

{ 18 , 6 , 2 , 2 3 , 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.

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 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.

{ 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 = 9 6 = 1.5

Substitute the common ratio into the recursive formula for geometric sequences and define a 1 .

a n = r a n 1 a n = 1.5 a n 1  for  n 2 a 1 = 6
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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.

{ 2 4 3 8 9 16 27 ... }

a 1 = 2 a n = 2 3 a n 1  for  n 2

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 · 2 n 1

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

Graph of the geometric sequence.

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.

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

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

a 2 = 2 a 1 = 2 ( 3 ) = 6
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





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