Chapter 9, Problem 4RE

a.

To determine

To find the common ratio.

Answer to Problem 4RE

The common ratio is $\text{'}-4\text{'}$ .

Explanation of Solution

Given information:

Given sequence is,

  $\frac{1}{2},-2,8,-\mathrm{32.......}$ Is geometric.

CALCULATION:

Given sequence is,

  $\frac{1}{2},-2,8,-\mathrm{32.......}$ Is geometric

To find the common ratio is the amount we multiply by each time in a geometric sequence.

Between the all consecutive terms.

2^nd term / 1^st term $=\frac{-2}{\frac{1}{2}}=-4$

3^rd term / 2^nd term $=\frac{8}{-2}=-4$

4^th term / 3^rd term $=\frac{-32}{8}=-4$

So, each number is $\text{'}-4\text{'}$ times the number before it, so the common ratio is $\text{'}-4\text{'}$ .

Hence,

The common ratio is $\text{'}-4\text{'}$ .

b.

To determine

To find the 7^th term.

Answer to Problem 4RE

The 7^th term is $-8192.$

Explanation of Solution

Given information:

Given sequence is,

  $\frac{1}{2},-2,8,-\mathrm{32.......}$ Is geometric.

CALCULATION:

Given sequence is,

  $\frac{1}{2},-2,8,-\mathrm{32.......}$ Is geometric

To find the 7^th term of the above given sequence first find the $n-th$ term of the above given sequence.

  $\begin{array}{l}an=a{\left(r\right)}^{n-1}\left[a=\text{first term}\text{.}n=\text{natural number}\text{.}r=\text{common ratio}\right]\\ a_7=a_1{\left(-4\right)}^{7-1}\left[n=7,a_1=\frac{1}{2},r=-4\right]\\ a_7=-8192\end{array}$

Hence,

The 7^th term is $-8192.$

c.

To determine

An explicit rule for $nth$ term.

Answer to Problem 4RE

An explicit rule for $nth$ term is $\frac{1}{8}{4}^x$

Explanation of Solution

Given information:

Given sequence is,

  $\frac{1}{2},-2,8,-\mathrm{32.......}$ Is geometric

CALCULATION:

Given sequence is,

  $\frac{1}{2},-2,8,-\mathrm{32.......}$ Is geometric

To find the $nth$ term of the above given sequence is defined explicit by,

  $\begin{array}{l}{a}_x=d{r}^{n-1}\left[r=\text{common difference}\text{. }a=1^{st}\text{term}.an=n-th\text{ term}\text{.}\right]\\ a_x=\frac{1}{2}{\left(-4\right)}^{n-1}\\ a_x=\frac{1}{8}{4}^x\end{array}$

Hence,

An explicit rule for $n-th$ term is $\frac{1}{8}{4}^x$