Chapter 8, Problem 51RE

(a)

To determine

To find: The first term of the geometric sequence.

Answer to Problem 51RE

The first term of the geometric sequence is $-6$ .

Explanation of Solution

Given information: Second and fifth term of the geometric sequence are $-3$ and $-\frac{3}{8}$ respectively.

Formula used :

  $n^{\text{th}}$ term of the geometric sequence (GP) $a,ar,a{r}^2,a{r}^3,\dots$ is given by $a_n=a{r}^{n-1}$ , where $a$ and $r$ denote the first term and the common ratio

of the geometric sequence respectively.

Calculation:

Substitute $n=2$ into $a_n=a{r}^{n-1}$ and simplify.

  $\begin{array}{c}{a}_2=a{r}^{2-1}\\ =a{r}^1\\ =ar\end{array}$

Since the second term is $-3$ , substitute $a_2=-3$ into the left side of the obtained equation. Label the resulting equation as equation (1).

  $-3=ar\dots \left(1\right)$

Now, Substitute $n=5$ into $a_n=a{r}^{n-1}$ and simplify.

  $\begin{array}{c}{a}_5=a{r}^{5-1}\\ =a{r}^4\end{array}$

Since the fifth term is $-\frac{3}{8}$ , substitute $a_5=-\frac{3}{8}$ into the left side of the obtained equation. Label the resulting equation as equation (2).

  $-\frac{3}{8}=a{r}^4\dots \left(1\right)$

Divide equation (1) by equation (2) and solve for $r$ .

  $\begin{array}{c}\frac{\left(-3\right)}{\left(-\frac{3}{8}\right)}=\frac{ar}{a{r}^4}\\ 8=\frac{1}{r^3}\\ r^3=\frac{1}{8}\\ r=\frac{1}{2}\end{array}$

Substitute $r=\frac{1}{2}$ into equation (1) and solve for a to obtain the required result.

  $\begin{array}{c}-3=a\left(\frac{1}{2}\right)\\ a=-6\end{array}$

Therefore, the first term of the geometric sequence is $a=-6$ .

(b)

To determine

To find: The common ratio presented in the given sequence.

Answer to Problem 51RE

The common ratio is $\frac{1}{2}$ .

Explanation of Solution

Given information: Second and fifth term of the geometric sequence are $-3$ and $-\frac{3}{8}$ respectively.

Formula used :

  $n^{\text{th}}$ term of the geometric sequence (GP) $a,ar,a{r}^2,a{r}^3,\dots$ is given by $a_n=a{r}^{n-1}$ , where $a$ and $r$ denote the first term and the common ratio of the geometric sequence respectively.

Calculation:

Substitute $n=2$ into $a_n=a{r}^{n-1}$ and simplify.

  $\begin{array}{c}{a}_2=a{r}^{2-1}\\ =a{r}^1\\ =ar\end{array}$

Since the second term is $-3$ , substitute $a_2=-3$ into the left side of the obtained equation. Label the resulting equation as equation (1).

  $-3=ar\dots \left(1\right)$

Now, Substitute $n=5$ into $a_n=a{r}^{n-1}$ and simplify.

  $\begin{array}{c}{a}_5=a{r}^{5-1}\\ =a{r}^4\end{array}$

Since the fifth term is $-\frac{3}{8}$ , substitute $a_5=-\frac{3}{8}$ into the left side of the obtained equation. Label the resulting equation as equation (2).

  $-\frac{3}{8}=a{r}^4\dots \left(1\right)$

Divide equation (1) by equation (2) and solve for $r$ .

  $\begin{array}{c}\frac{\left(-3\right)}{\left(-\frac{3}{8}\right)}=\frac{ar}{a{r}^4}\\ 8=\frac{1}{r^3}\\ r^3=\frac{1}{8}\\ r=\frac{1}{2}\end{array}$

Therefore, the common ratio is $r=\frac{1}{2}$ .

(c)

To determine

To find: An explicit formula for the $n^{\text{th}}$ term.

Answer to Problem 51RE

An explicit formula for $n^{\text{th}}$ term is $-3{\left(2\right)}^{2-n}$ .

Explanation of Solution

Given information: Second and fifth term of the geometric sequence are $-3$ and $-\frac{3}{8}$ respectively.

Formula used :

  $n^{\text{th}}$ term of the geometric sequence (GP) $a,ar,a{r}^2,a{r}^3,\dots$ is given by $a_n=a{r}^{n-1}$ , where $a$ and $r$ denote the first term and the common ratio of the geometric sequence respectively.

Calculation:

From part (a) and part (b) the first term and the common term are $a=-6$ and $r=\frac{1}{2}$ respectively.

Substitute $a=-6$ and $r=\frac{1}{2}$ into $a_n=a{r}^{n-1}$ and simplify to obtain the required result.

  $\begin{array}{c}{a}_n=-6{\left(\frac{1}{2}\right)}^{n-1}\\ =-\left(3\times 2\right){\left(\frac{1}{2}\right)}^{n-1}\\ =-\left(3\times 2\right){\left(2\right)}^{1-n}\\ =-3{\left(2\right)}^{1+1-n}\\ =-3{\left(2\right)}^{2-n}\end{array}$

Therefore, explicit formula for the given problem is $-3{\left(2\right)}^{2-n}$ .