Chapter 7.3, Problem 6MRPS

Solution

To calculate : The rule for the height of top of $n^{th}$ stair, height of the top of ${10}^{th}$ stair and explain how to modify the rule so that it gives height of the bottom $n^{th}$ stair if a builder is constructing a staircase for a deck. At the foot of the staircase, there is a concrete slab 2 inches tall, and each stair is 7 inches tall.

The rule for the height of top of $n^{th}$ stair is $a_n=2+7n$ , height of the top of ${10}^{th}$ stair is 72 inches and the modified rule that gives height of the bottom $n^{th}$ stair is $a_n=7n-5$ .

Given information :

A builder is constructing a staircase for a deck. At the foot of the staircase, there is a concrete slab 2 inches tall, and each stair is 7 inches tall.

Formula used :

The $n^{th}$ term of a arithmetic series with first term as $a_1$ and common difference $d$ is given by $a_n=a_1+\left(n-1\right)d$ .

The sum of first $n$ terms of an arithmetic series is $S_n=n\left(\frac{a_1+a_2}{2}\right)$ .

Calculation :

It can be observed that the height of the first stair is $7+2=9$ in that is $a_1=9$ .

The difference in height of successive stairs is 7 inches, so $d=7$ .

Substitute all the values into the formula $a_n=a_1+\left(n-1\right)d$ to calculate the value of $a_n$ .

  $\begin{array}{c}{a}_n=9+7\left(n-1\right)\\ =9+7n-7\\ a_n=2+7n\end{array}$

Substitute the value of $n$ as 10 to calculate the height of the top of ${10}^{th}$ stair.

  $\begin{array}{c}{a}_{10}=2+7\left(10\right)\\ =2+70\\ =72\end{array}$

Now, to calculate the height of the bottom of $n^{th}$ stair, subtract the height of $n^{th}$ stair from $a_n$ .

  $\begin{array}{c}{a}_n-7=2+7n-7\\ =7n-5\end{array}$

Thus, the rule for the height of top of $n^{th}$ stair is $a_n=2+7n$ , height of the top of ${10}^{th}$ stair is 72 inches and the modified rule that gives height of the bottom $n^{th}$ stair is $a_n=7n-5$ .