Chapter 7.2, Problem 65PS

Solution

i.

To identify: The rule for the number of visible blocks in the nth layer where n=1 represents the front layer.

Sol LeWitt's sculpture Four-Sided Pyramid in the National Gallery of Art Sculpture Garden is made of concrete blocks. As shown in the diagram, each layer has 8 more visible blocks than the layer in front of it.

  

The required rule is $a_n=8n-4$ .

Given information:

Given figure is:

  

Explanation:

Consider the given sequence.

The general formula for an arithmetic sequence is $a_n=a_1+\left(n-1\right)d$ .

Fist note that the differences of visible blocks between two layers are the same, so it the numbers of visible blocks in layers form an arithmetic sequence.

Each layer has $8$ more visible blocks, thus the common difference is $d=8$ .

The number of visible blocks in the first layer is $4$ , thus the first term is

  $a_1=4$

Now, let's determine the $nth$ term of this sequence.

Put the value of $a_1=4\text{and}d=8$ into $a_n=a_1+\left(n-1\right)d$

  $\begin{array}{l}{a}_n=4+\left(n-1\right)8\\ a_n=4+8n-8\\ a_n=8n-4\end{array}$

ii.

To calculate: How many of the pyramid's blocks are visible? When you view the pyramid from one corner, a total of 12 layers are visible.

  

Therefore, there are $576$ visible block

Given information:

The given terms are $a_1=4,n=12\text{and}d=8$ .

Calculation:

Consider the given sequence.

It is given that $a_1=4,n=12\text{and}d=8$

Put the value of $n=12$ into $a_n=8n-4$

  $\begin{array}{l}{a}_n=8n-4\\ a_{12}=8\left(12\right)-4\\ a_{12}=92\end{array}$

The general sum formula for a arithmetic sequence is $s_n=\frac{n\left(a_1+a_9\right)}{2}$ .

Put the value of $a_1=4,n=12\text{and}{a}_{12}=92$ into $s_n=\frac{n\left(a_1+a_9\right)}{2}$

  $\begin{array}{l}{s}_{12}=\frac{12\left(4+92\right)}{2}\\ s_{12}=\frac{12\cdot 96}{2}\\ s_{12}=576\end{array}$

Therefore, there are $576$ visible block