Question

The second term in a geometric sequence is 20. The fourth term in the same sequence is 11.25. What is the common ratio

Answer 1

let
`a_n` be the n'th term of the sequence.

so
`a_1` is the first term,
`a_2` the second term and so on...

In a geometric sequence with
`a_1=c`, and common ratio r, the terms are as follows:


`a_1=c`

`a_2=cr`

`a_3=c r^(2)`

`a_4=c r^(3)`
.
.
that is, each term is its previous term times the common ratio r.

In our example


`a_2=cr=20` and
`a_4=c r^(3)=11.25`


`(a_2)/(a_4)= (cr)/(c r^(3))= (1)/( r^(2))= (20)/(11.25)= 1.778`

so
`r^(2) =1/1.778=0.56`


`r= √(0.56)= 0.75`


Answer: r=0.75