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:

$f(n)=1000\left(\dfrac{1}{4}\right)^{n-1}$

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,

$f(n)=1000\left(\dfrac{1}{4}\right)^{n-1}$         ...(ii)

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

$a=1000$

$f(1)=1000$

And,

$r=\dfrac{1}{4}$

For $n=1$,

$f(1)=1000\left(\dfrac{1}{4}\right)^{1-1}$

$f(1)=1000\left(\dfrac{1}{4}\right)^{0}$

$f(1)=1000(1)$

$f(1)=1000$

For $n=2$,

$f(2)=1000\left(\dfrac{1}{4}\right)^{2-1}$

$f(2)=1000\left(\dfrac{1}{4}\right)^{1}$

$f(2)=1000\times \dfrac{1}{4}$

$f(2)=250$

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

$f(3)=62.5$

$f(4)=15.625$

$f(5)=3.90625$

Therefore, $f(1)=1000,r=\dfrac{1}{4}$ and the first five terms are $1000,250,62.5,15.625,3.90625$.

Answer 2

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 .