What is the 7th term of a geometric sequence whose third term is 27 and whose common ratio is 3

Índice Show

Geometric Sequences
Geometric Series
The Sum to Infinity of a Geometric Sequence

Check out the new look and enjoy easier access to your favorite features

Page 2

If a sequence is geometric there are ways to find the sum of the first n terms, denoted Sn, without actually adding all of the terms.

To find the sum of the first Sn terms of a geometric sequence use the formula
Sn=a1(1−rn)1−r,r≠1,
where n is the number of terms, a1 is the first term and r is the common ratio.

The sum of the first n terms of a geometric sequence is called geometric series.

Example 1:

Find the sum of the first 8 terms of the geometric series if a1=1 and r=2.

S8=1(1−28)1−2=255

Example 2:

Find S10 of the geometric sequence 24,12,6,⋯.

First, find r.

r=r2r1=1224=12

Now, find the sum:

S10=24(1−(12)10)1−12=306964

Example 3:

Evaluate.

∑n=1103(−2)n−1

(You are finding S10 for the series 3−6+12−24+⋯, whose common ratio is −2.)

Sn=a1(1−rn)1−rS10=3[1−(−2)10]1−(−2)=3(1−1024)3=−1023

In order for an infinite geometric series to have a sum, the common ratio r must be between −1 and 1. Then as n increases, rn gets closer and closer to 0. To find the sum of an infinite geometric series having ratios with an absolute value less than one, use the formula, S=a11−r, where a1 is the first term and r is the common ratio.

Example 4:

Find the sum of the infinite geometric sequence
27,18,12,8,⋯.

First find r:

r=a2a1=1827=23

Then find the sum:

S=a11−r

S=271−23=81

Example 5:

Find the sum of the infinite geometric sequence
8,12,18,27,⋯ if it exists.

First find r:

r=a2a1=128=32

Since r=32 is not less than one the series has no sum.

There is a formula to calculate the nth term of an geometric series, that is, the sum of the first n terms of an geometric sequence.

See also: sigma notation of a series and sum of the first n terms of an arithmetic sequence

In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation.

An geometric sequence or progression, is a sequence where each term is calculated by multiplying the previous term by a fixed number.

Geometric Sequences

This fixed number is called the common ratio, r.
The common ratio can be positive or negative, an integer or a fraction.

The common ratio can be calculated by dividing any term by the one before it.
e.g. Common ratio = t n+1 ÷ t n

The first term of a geometric sequence is shown by the variable a.

Geometric Sequence	First term, a	Common ratio, r
2, 6, 18, 54, ...	2
20, 10, 5, 2.5, ...	20	10 ÷ 20 = 0.5 5 ÷ 10 = 0.5 etc.

General Term, tn

A geometric sequence can be written:

First term

Second term

Third term

Fourth term

General term (n th term)

t 1

t 2

t 3

t 4

...

t n

ar2

ar3

...

ar n − 1

Example 1

What is the common ratio of the geometric sequence:

20, 10, 5, 2.5,, ...

Common ratio = t n+1 ÷ t n

Second term ÷ first term = t2 ÷ t1 = 10 ÷ 20 = 0.5

Check:

Third term ÷ second term = 5 ÷ 10 = 0.5

The common ratio is 0.5

Example 2

Find the 8th term of the geometric sequence:

3, 9, 27, 81, ...

Common ratio, r = 9 ÷ 3 = 3
First term a = 3

using tn = ar n − 1

t8 = 3 x 3 8- 1
= 3 x 3 7
= 6561

The 8th term is 6561

Example 3

Which term of the sequence 2, 4, 8, 16, ... would be equal to 1024?

Common ratio, r = 2
First term, a = 2

Using tn = a r n-1

1024 =2 x 2 n-1
1024 = 21 x 2 n − 1
1024 = 2 n
n = 10

1024 is the 10th term.

Example 4

The third term of an geometric sequence of positive terms is 8 and the fifth term is 32.

Find the first term, a, and the common ratio, r, and thus list the first four terms of the sequence.

t 3 = 8
t 5 = 32

using tn = a r n-1

32 = ar 5 − 1
8 = ar 3 − 1

32 = ar 4
8 = ar 2

4 = r2 ( dividing)
r = ±2

The common ratio is 2 (discard -2)

8 = a x 22
a = 2 the first term

The sequence is 2, 4, 8, 16, ...

Geometric Series

If terms of a geometric sequence are added together a geometric series is formed.

2 + 4 + 8 + 16 is a finite geometric series
2 + 4 + 8 + 16 + ... is an infinte geometric series

To find the sum of the first n terms of a geometric sequence use the formula:

Sum of first n terms of a geometric sequence

wherer = common ratioa = first term

n = number of terms

If the common ratio is a fraction i.e. -1 < r < 1 then an equivalent formula, shown below is easier to use.

Sum of first n terms of a geometric sequence

for when -1 < r < 1 i.e. r is a fraction

Example What is the sum of the first 10 terms of the geometric sequence: 3, 6, 12, ...

Common ratio	r = 6 ÷ 3 = 2
Number of terms	n = 10
First term	a = 3

The Sum to Infinity of a Geometric Sequence

Spreadsheets are very useful for generating sequences and series.

For a geometric sequence with a common ratio greater than 1:

The formula in cell B3 is = B2*2

The formula in cell D3 is =D2 + B3

The fill down command is then used to complete the sequences.

It can be seen that as successive terms are added the sum of the terms increases.
If there were an infinite number of terms the sum would be infinity.

For a geometric sequence with a common ratio less than 1:

The formula in cell B3 is = B2*2

The formula in cell D3 is =D2 + B3

The fill down command is then used to comlete the sequences.

It can be seen that as successive terms are added the sum of the terms appears to be heading towards 16.
If there were an infinite number of terms the sum would be 16.

This is called the sum to infinity of a geometric sequence and only applies when the common ratio is a fraction

i.e. -1 < r < +1.
r can be positive or negative.

The following formula can be used:

Sum to infinity of geometric sequence

wherer = common ratio

a = first term

Example

Find the sum to infinity of the geometric sequence 8, 4, 2, 1, ...