Chapter 6.5, Problem 39E

a.

To determine

To write a function that represents the sequence of figures

Answer to Problem 39E

The function that represents the sequence of figures is $f(n)=9^{n-1}$

Explanation of Solution

Given: The $\text{First term, }{a}_1=1$ and $\text{Common ratio, r = 9}$

  

The function for a geometric sequence is: $f(n)=a_1{r}^{n-1}$

  $a_1=1$ , $\text{r = 9}$

Now,

  $\begin{array}{l}f(n)=a_1{r}^{n-1}\\ $ f(n)=1{(9)}^{n-1}$f(n)=9^{n-1}\end{array}$

Hence,

The function that represents the sequence of figures is $f(n)=9^{n-1}$

b.

To determine

To describe the 10^th figure in the sequence.

Answer to Problem 39E

The $10th$ figure of the sequence will contain $9^9$ squares with equal sides.

Explanation of Solution

Given: The $\text{First term, }{a}_1=1$ and $\text{Common ratio, r = 9}$

  

The function for a geometric sequence is: $f(n)=a_1{r}^{n-1}$

  $a_1=1$ , $\text{r = 9}$

Now,

  $\begin{array}{l}f(n)=a_1{r}^{n-1}\\ $ f(n)=1{(9)}^{n-1}$f(n)=9^{n-1}\end{array}$

Therefore, For the tenth figure of the sequences, $n=10$

  $\begin{array}{l}f(10)=9^{10-1}\\ f(10)=9^9\end{array}$

Hence,

The $10th$ figure of the sequence will contain $9^9$ squares with equal sides.