Chapter MPS, Problem 11.6MPS

a.

To determine

Write a recursive formula for the height of each floor.

Answer to Problem 11.6MPS

  $a_{n+1}=13+a_n,a_1=9$

Explanation of Solution

Given information:

A $25$ -story building has a basement that is $4$ feet below street level. Each floor is $13$ feet high.

Calculation:

So, the height of first floor in feet from street level is,

  $\begin{array}{l}{a}_1=-4+13\\ =9\end{array}$

That is, $a_1=9$ .

Now, the second floor has height $a_2=13+a_1$ and height of third floor is $a_3=13+a_2$ .

So, the recursive formula for the height of $n$ the floor is,

  $a_{n+1}=13+a_n,$ here $a_1=9$

Hence, the recursive formula for the height of each floor is $a_{n+1}=13+a_n,a_1=9$ .

b.

To determine

Find the height of the first $5$ floors.

Answer to Problem 11.6MPS

  $61feet$

Explanation of Solution

Given information:

A $25$ -story building has a basement that is $4$ feet below street level. Each floor is $13$ feet high.

Calculation:

The recursive formula for the height of each floor is $a_{n+1}=13+a_n,a_1=9$ .

Then, substitute $n=4$ in the formula,

  $\begin{array}{l}{a}_{n+1}=13+a_n\\ a_5=13+a_4\end{array}$

Find the values of $a_2,a_3,and{a}_4$ as follows:

  $\begin{array}{l}{a}_2=13+a_1\\ a_2=13+9\\ a_2=22\end{array}$

  $\begin{array}{l}{a}_3=13+a_2\\ a_3=13+22\\ a_3=35\\ a_4=13+a_3\\ a_4=13+35\\ a_4=48\end{array}$

Substitute the value $a_4=48$ in $a_5=13+a_4$ .

  $\begin{array}{l}{a}_5=13+48\\ =61\end{array}$

Hence, the height of first $5$ floors from street level is $61feet$ .