Chapter 6.6, Problem 37E

To determine

To graph first four terms of the sequence with given description

Also write recursive and explicit rule

Answer to Problem 37E

First four terms of sequence are $-1,3,-9,27$

Recursive rule of the sequence

  $a_n=-3a_{n-1}$

Explicit rule of the sequence

  $a_n=\left(-1\right){\left(-3\right)}^{n-1}$

Graph of the sequence plotting points as $\left(n,a_n\right)$

  

Explanation of Solution

Given information:

The first term of the sequence is16. Each term of the sequence is half the preceding term.

Concept used:

Recursive equation for an arithmetic sequence is

  $a_n=a_{n-1}+d\text{ (where d is common difference)}$

Recursive equation for a geometric sequence is

  $a_n=r.a_{n-1}\text{ (where r is common ratio)}$

Equation of explicit rule for arithmetic sequence is given by

  $a_n=a_1+\left(n-1\right)d$ (where d is the common difference)

Equation of explicit rule for geometric sequence is given by

  $a_n=a_1{r}^{n-1}$ (where r is the common ratio)

Given first term of the sequence is $-1$

  $\left(n,a_n\right)$

This means $a_1=-1$

Also given each term of the sequence is half the preceding term.

This means common ratio, $r=-3$

Therefore,

Other three terms of sequence are

  $\begin{array}{l}{a}_2=a_1\times r=-1\times -3=3\\ a_3=a_2\times r=3\times -3=-9\\ a_4=a_3\times r=-9\times -3=27\end{array}$

Hence,

First four terms of sequence are $-1,3,-9,27$

Graph of this sequence is given below (plot points $\left(n,\left(a_n\right)\right)$ to obtain the graph)

  

The sequence is a discrete function. So, the points are not connected by line

Recursive equation

  $a_n=-3a_{n-1}$

Explicit equation

  $a_n=\left(-1\right){\left(-3\right)}^{n-1}$