What is the 7th term of the geometric sequence?

A geometric sequence has a constant ratio (common ratio) between consecutive terms.

For 3, 9, 27, ... the common ratio is 3 because:
3 X 3 = 9
9 X 3 = 27

So to find the 7th term you can do it two ways:

One way:
3 is the 1st term, 9 is the 2nd term, 27 is the 3rd term so then
4th term: 27 X 3 = 81
5th term: 81 X 3 = 243
6th term: 243 X 3 = 729
7th term: 729 X 3 = 2,187

Another way:
You can use the explicit formula #a_n = a_1 *r^(n-1)#, where #a_n# is the nth term, #a_1# is the first term, n is the number of the term, and r is the common ratio

so #a_7 = 3 * 3^ (7-1)#
#a_7 = 3 * 3^ (6)#
#a_7 = 3 * 729#
#a_7 = 2,187#

Both ways get you to the same answer that the 7th term in that geometric sequence is 2, 187 .

Solution:

The nth term of the geometric sequence is given by, an = a rn - 1,

Where a and r being the first term and the common ratio respectively.

Given a1 = a = 128 and a3 = 8

We know an = a rn - 1,

⇒ a3 = a.r3 - 1

⇒ 8 = 128 × r2

⇒ r2 = 8/128

⇒ r2 = 1/16

⇒ r = ±1/4

Now, a7 = a.r7 - 1 = a.r6

⇒ a7 = 128 × (1/4)6

⇒ a7 = 128/4096

⇒ a7 = 1/32

If r = -1/4,

⇒ a7 = 128 × (-1/4)6

⇒ a7 = 128/4096

⇒ a7 = 1/32

Therefore, the 7th term of the geometric sequence a7 is 1/32.

What is the 7th term of the geometric sequence where a1 = 128 and a3 = 8?

Summary:

The 7th term of the geometric sequence where a1 = 128 and a3 = 8 is 1/32.

Algebra Examples

Popular Problems

Algebra

Find the 7th Term 1 , 2 , 4 , 8 , 16 , 32

, , , , ,

Step 1

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by gives the next term. In other words, .

Geometric Sequence:

Step 2

This is the form of a geometric sequence.

Step 3

Substitute in the values of and .

Step 4

Multiply by .

Step 5

Substitute in the value of to find the th term.

Step 6

Subtract from .

Step 7

Raise to the power of .

You can put this solution on YOUR website!
.

Let "a" be the first term of the progression and "r" be the common ratio.

Then   =   and   =  .

Therefore,   =  = 2,  and,  hence,   = .


Then the sum is

254 = 
 = 


    =  = 


    =  = 


    = .


Hence,  a = .


Answer.  The first term is  a = .

On geometric progressions, see the lessons
    - Geometric progressions
    - The proofs of the formulas for geometric progressions
    - Problems on geometric progressions
    - Word problems on geometric progressions
    - One characteristic property of geometric progressions
    - Solved problems on geometric progressions
    - Fresh, sweet and crispy problem on arithmetic and geometric progressions

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic

What is the 7th term of the geometric sequence 1?

What is the 7th term of the geometric sequence where a1 1024 and a4 =

What is the 7th term of the geometric sequence 3 6 12?