Chapter 9.1, Problem 4QR

(a)

To determine

To find: The common ratio in the given sequence.

Answer to Problem 4QR

The common difference is $-\frac{1}{2}$.

Explanation of Solution

Given information: $\left\{8,-4,2,-1,\cdots \right\}$

Concept used: The division of any term (except the first term) with the previous term is known as common ratio.

Calculation:

Divide 6 by 2 to obtain the common ratio, $r$ .

  $\begin{array}{c}r=\frac{-4}{8}\\ =-\frac{1}{2}\end{array}$

(b)

To determine

To find: The tenth term of the given sequence.

Answer to Problem 4QR

The tenth term is $-\frac{1}{64}$.

Explanation of Solution

Given information: $\left\{8,-4,2,-1,\cdots \right\}$

Concept used: The $n$ -th term of the sequence is obtained by the formula, $a_n=a{r}^{n-1}$ , where $a$ is the initial term and $r$ is the common ratio.

Calculation:

Substitute $n=10$ into the formula, $a_n=a{r}^{n-1}$ , and simplify it to determine the value of the tenth term. Note that $a=8$ and $r=-\frac{1}{2}$ .

  $\begin{array}{c}{a}_{10}=8{\left(-\frac{1}{2}\right)}^{10-1}\\ =-\frac{1}{64}\end{array}$

(c)

To determine

To find: A rule for the $n$ -th term.

Answer to Problem 4QR

The rule for n-th term is $a_n=8{\left(-\frac{1}{2}\right)}^{n-1}$.

Explanation of Solution

Given information: $\left\{8,-4,2,-1,\cdots \right\}$

Concept used: The $n$ -th term of the sequence is obtained by the formula, $a_n=a{r}^{n-1}$ , where $a$ is the initial term and $r$ is the common ratio.

Calculation:

Substitute $a=8$ and $r=-\frac{1}{2}$ into the formula, $a_n=a{r}^{n-1}$ , and simplify it to determine the required rule.

  $a_n=8{\left(-\frac{1}{2}\right)}^{n-1}$