Chapter 7.8, Problem 40PPE

To determine

To Write: Write the explicit and recursive formula if given sequence is a geometric sequence.

Answer to Problem 40PPE

It is a geometric sequence. The recursive formula is $a_n=a_{n-1}.\left(-3\right);a_1=3$

and explicit formula is $a_n=3.{\left(-3\right)}^{n-1}$

Explanation of Solution

Given information : The given sequence is $3,-9,27,-81,\mathrm{...}$

We have the given sequence is $3,-9,27,-81,\mathrm{...}$

A sequence is said to be geometric sequence if each term is obtained by multiplying the common ratio to the previous term.

Here the first term is $a_1=3$ and common ratio = $\frac{-9}{3}=-3$

Now, $a_2=a_{1}r=3\times -3=-9$

  $a_3=a_{2}r=-9\times -3=27$ and so on

Therefore, the given sequence is a geometric sequence.

Now, we have

The recursive formula is

  $a_n=a_{n-1}.r;a_1=3$

Now, substituting the value of r

  $a_n=a_{n-1}.\left(-3\right);a_1=3$

The explicit formula is

  $a_n=a_1{r}^{n-1}$

Substituting the value of a and r

  $a_n=3.{\left(-3\right)}^{n-1}$