<< 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

find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
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
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