Question

How do you write a rule for the nth term of the geometric term given the two terms `a_2=2, a_5=1/4`?

Answer 1

`a_n=4*(1/2)^(n-1)`

Explanation:

The general form for a geometric sequence is:
`a_n=a_1*r^(n-1)`

With this, you can write a system of equations:
`a_2=a_1*r^(2-1)`
`a_5=a_1*r^(5-1)`

`2=a_1*r^1`
`0.25=a_1*r^4`
`r^4/r^1=0.25/2`
`r^3=0.125`
`r=0.5`

Now, find the first term.
`a_2=a_1*0.5^1`
`2=a_1*0.5`
`a_1=4`

Using the formula stated above, we get:
`a_n=4*0.5^(n-1)`