Chapter 4.6, Problem 46E

(a)

To determine

To write: a function that represents the arithmetic sequence.

Answer to Problem 46E

The function that represents the arithmetic sequence is $f\left(n\right)=-8n+64$ .

Explanation of Solution

Given:

  

Calculation:

The first term is 56, and the common difference is -8.

Consider the function for the arithmetic sequence

  $f\left(n\right)=a_1+\left(n-1\right)d$

Put the value $a_1=56$ and $d=8$

  $\begin{array}{l}f\left(n\right)=56+\left(n-1\right)\left(-8\right)\hfill \\ =56-8n+8\hfill \\ =-8n+64\hfill \end{array}$

Conclusion:

Therefore, the function that represents the arithmetic sequence is $f\left(n\right)=-8n+64$ .

(b)

To determine

To find: the number of weeks in the movie earn $16 million.

Answer to Problem 46E

The movie earns $16 million in 6 weeks.

Explanation of Solution

Calculation:

Now find the value of n for which $f\left(n\right)=16$

So,

  $\begin{array}{l}f\left(n\right)=16\hfill \\ -8n+64=16\hfill \end{array}$

Subtract by 64 on both sides

  $-8n=-48$

Divide by -8 on both sides

  $n=6$

Conclusion:

Therefore, the movie earns $16 million in 6 weeks.

(c)

To determine

To find: the overall amount that the movie earns overall.

Answer to Problem 46E

The movie earns overall $224 million of dollar.

Explanation of Solution

Calculation:

Now find the value of n for which $f\left(n\right)=0$

So,

  $\begin{array}{l}f\left(n\right)=0\\ -8n+64=0\end{array}$

Subtract by 64 on both sides

  $-8n=-64$

Divide by -8 on both sides

  $n=8$

Thus, the movie earn money for 8 weeks

  $\begin{array}{l}{a}_5=a_4+d\hfill \\ =32-8\hfill \\ =24\hfill \end{array}$

And

  $\begin{array}{l}{a}_7=a_6+d\hfill \\ =16-8\hfill \\ =8\hfill \end{array}$

Now, sum for the first 8 week

  $\begin{array}{l}\text{Total}=a_1+a_2+a_3+a_4+a_5+a_6+a_7+a_8\hfill \\ \begin{array}{l}=\text{ 56}+48+40+32+24+16+8+0\\ =224\end{array}\hfill \end{array}$

Conclusion:

Therefore, the movie earns overall $224 million of dollar.