Question

A geometric sequence has only positive terms. The third term is 6 and the fifth term is 150. Find the common ratio. [Answer format: integer, no units

Answer 1

Hi,

Explanation:

In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio (r).

We are given the third term (
`a_3 = 6`) and the fifth term (
`a_5 = 150`).

We can use these to set up equations.

The general formula for the nth term of a geometric sequence is given by:

`a_n=a_1*r^(n-1)`

where:

• 
  `a_n`​ is the nth term
• 
  `a_1`​ is the first term
• r is the common ratio

Given that
`a_3=6` and
`a_5=150`, we can set up the following equations:

`\left\{\begin{array}{ccc}a_3=a_1*r^2\\a_5=a_1*r^4\\\end {array} \right.\\\\\\\left\{\begin{array}{ccc}6=a_1*r^2\\150=a_1*r^4\\\end {array} \right.\\`

Dividing the second equation by the first gives:

`25=(a_1*r^4)/(a_1*r^2) =r^2\\r=5\ or\ r=-5\\`

Since all terms are positive, the ration is positive:

So, the common ratio r is 5