Question

Write an explicit rule in function form for the sequence represented by the given terms.

a. f(5)=3 and f(7)=147

b. f(3)=10 and f(5) =1440

Answer 1

The general relation about a geoemtric sequence is

`a_(n)=a_(1)r^(n-1)`

For the first sequence, we have

`3=a_(1)r^(5-1)`

`147=a_(1)r^(7-1)`

Which is a system of equations. We can isolate a variable in the first equation to replace that expression into the second equation.

`a_(1)=(3)/(r^(4) )`

`147=(3)/(r^(4) )r^(6)`

`r^(2)=(147)/(3)\\ r=\sqrt[2]{49}\\r=7`

Now, we replace this value to find the other one

`3=a_(1)(7)^(4)\\ a_(1)=(3)/(2401)`

Therefore, the explicit rule function is

`a_(n)=(3)/(2401) * (7)^(n-1)`

Now, we use the same process for the second sequence.

`10=a_(1)r^(3-1) \\a_(1)=(10)/(r^(2) )`

The second equation is

`1440=a_(1)r^(5-1)\\a_(1)=(1440)/(r^(4) )`

Now, we solve the following expression

`(10)/(r^(2) )=(1440)/(r^(4) )`

We solve for
`r`

`(r^(4) )/(r^(2) )=(1440)/(10)\\r^(2)=144\\ r=√(144) \\r=12`

Then

`a_(1)=(10)/((12)^(2) ) =(10)/(144)=(5)/(72)`

Therefore, the function that models the second sequence is

`a_(n)=(5)/(72) * (12)^(n-1)`

Notice that
`a_(n)` is the dependent variable and
`n` is the independent variable.