<< 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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana 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