Chapter 8, Problem 1CR

To determine

Find the pattern and write a rule for the $n\text{th}$ layer using given figure.

Answer to Problem 1CR

The rule for the $n\text{th}$ layer is $a_n=n^2+n$ .

Explanation of Solution

Given:

The given figure is

  

Calculation:

Find the number of cubes in each layer.

  

In first layer $\to$$1\cdot 2=2$

In second layer $\to$$2\cdot 3=6$

In third layer $\to$$3\cdot 4=12$

In fourth layer $\to$$4\cdot 5=20$

The rule for the $n\text{th}$ layer is

  $\begin{array}{l}{a}_n=n\left(n+1\right)\\ \Rightarrow a_n=n^2+n\end{array}$

Hence the rule for the $n\text{th}$ layer is $a_n=n^2+n$ .