Question

Use the following explicit function to identify f(1) & r, and list the first five terms of the sequence.

f(n) = 1000(1/4)^n-1
f(1) =?
r = ?
then the 5 terms:
___, ___, ___, ___, ___...​

Answer 1

Given:

The explicit function is:

To find:

and first 5 terms.

Solution:

The explicit formula a geometric sequence is:

...(i)

Where, a is the first term and r is the common ratio.

We have,

...(ii)

On comparing (i) and (ii), we get

And,

For ,

For ,

Similarly, substituting , we get

Therefore, and the first five terms are .

Answer 2

Given:

The explicit function is:

`f(n)=1000\left((1)/(4)\right)^(n-1)`

To find:

`f(1)=?,r=?` and first 5 terms.

Solution:

The explicit formula a geometric sequence is:

`f(n)=ar^(n-1)` ...(i)

Where, a is the first term and r is the common ratio.

We have,

`f(n)=1000\left((1)/(4)\right)^(n-1)` ...(ii)

On comparing (i) and (ii), we get

`a=1000`

`f(1)=1000`

And,

`r=(1)/(4)`

For
`n=1`,

`f(1)=1000\left((1)/(4)\right)^(1-1)`

`f(1)=1000\left((1)/(4)\right)^(0)`

`f(1)=1000(1)`

`f(1)=1000`

For
`n=2`,

`f(2)=1000\left((1)/(4)\right)^(2-1)`

`f(2)=1000\left((1)/(4)\right)^(1)`

`f(2)=1000* (1)/(4)`

`f(2)=250`

Similarly, substituting
`n=3,4,5`, we get

`f(3)=62.5`

`f(4)=15.625`

`f(5)=3.90625`

Therefore,
`f(1)=1000,r=(1)/(4)` and the first five terms are
`1000,250,62.5,15.625,3.90625`.