If you have the geometric sequence 9, 3, 1, 1/3 what are the next three term?

Question

I have tried multiple times to figure it out But I was given the geometric sequence 9, 3, 1, 1/3 and I need the next three geometric terms.

Answer 1

A geometric sequence is created by multiplying a term by a common ratio to determine the next term.

a_n = a_n-1(r)

Here the common ratio, r, is 1/3.

 

9, 3, 1, 1/3, 1/9, 1/27, 1/81

Answer 2

In the geometric sequence, there is a pattern with how the numbers are aligned as well as predictability.

 

Notice how the numbers decrease. You have 9, then 3, then 1, etc. You can get from 9 to 3 by subtracting 6, but that does not work for 3 to 1 (3-6≠1). However, dividing by 3 does work, 9÷3=3 and 3÷3=1. It still works when getting to 1/3 (1 ÷3=1/3). The key thing is to know that a fraction is division and division is a fraction. It is the same thing!

 

So the pattern continues, 1/3 must by divided by 3. The trick is that you're dividing by a fraction so you multiply by its reciprocal ("flipped"). 1/3 ÷3 (1/3 ÷ 3/1) is 1/3 • 1/3=1/9. Moving on with multiplying by a reciprocal, 1/9 • 1/3=1/27 and 1/27 • 1/3=1/81.

 

So the next three terms are: 1/9, 1/27, 1/81