Chapter 8.3, Problem 37E

(a)

To determine

To find: the ${14}^{th}$ term of the geometric sequence using the table feature of a graphing utility.

Answer to Problem 37E

  ${14}^{th}$ term of the geometric sequence is $633.57$ .

Explanation of Solution

Given information:

Given geometric sequence

  $a_1=7,r=\sqrt{2}$

Calculation:

To have the $n^{th}$ term of the geometric sequence, firstly click on $2^{nd}$ button and then click on STAT button. After this, a new screen will appear. The screenshot is given as below:

  

Now, move to the OPS option and then move to the $5^{th}$ option and press the ENTER button. After this, give the input in this function. The pattern of the giving the input. Then finally press the ENTER button to have the desired value.

Therefore, the ${14}^{th}$ term is $633.57$ .

(b)

To determine

To find: the ${14}^{th}$ term of the geometric sequence algebraically.

Answer to Problem 37E

  ${14}^{th}$ term of the geometric sequence is $633.57$ .

Explanation of Solution

Concept used:

The $n$ th term of geometric sequence has the form $a_1=a_1{r}^{n-1}$ where $r$ is the common ratio of the consecutive terms of the sequence.

Every geometric sequence can be written in the following form.

  $a_1=a_1,a_2=a_{1}r,a_3=a_1{r}^2,a_4=a_1{r}^3,\mathrm{...},a_n=a_1{r}^{n-1},\mathrm{..}$

Calculation:

The ${14}^{th}$ term of this geometric sequence can be calculated by using formula:

  $a_{n+1}=a{r}^n$

So,

  $\begin{array}{c}{a}_{13+1}=7\times {\left(\sqrt{2}\right)}^{13}\\ a_{14}=7\times 2^6\sqrt{2}\\ a_{14}=633.5676759\\ \approx 633.57\end{array}$

Hence, ${14}^{th}$ term of this geometric sequenceis $633.57$ .