Question

Four positive numbers form a geometric sequence. The sum of the four numbers is one more then the common ratio. If the first number is 1/10, find the common ratio.

Answer 1

r=3 where r is the common ratio

Explanation:

The sum of geometric progression for n terms is given by:

`S_(n)=a_(1) ( 1-r^(n))/ 1-r`, where
`a_(1)` is the

first term of the series and r is the common ratio.

Now, according to the question, there are four terms and they form a geometric progression. So sum of four terms is given as
`S_(4)= 1+r` and
`a_(1)=1/10`.

Also according to the above formula:

`S_(4)=a_(1) ( 1-r^(4))/ 1-r`

Using the values as given in the question into the above equation we get:

`1+r =1/10 ( 1-r^(4))/ 1-r`

`(1+r)(1-r)=1/10 ( 1-r^(4))`

[ Using formula
`a^(2) -b^(2) =( a+b)(a-b)` ]

`(1-r^(2))=1/10 ( 1-r^(2))(1+r^(2))`

`10=(1+r^(2))`

`10-1=(r^(2))`

`√(9)=(r)`

`r=3` which is the required answer.