Chapter 7.1, Problem 67PS

Solution

a.

To state: The number of balls in the fifth layer if the number $a_n$ of balls per layer is given by $a_n=\frac{n\left(n+1\right)}{2}$ where $n=1$ represents the top layer.

The resultant balls are 15.

Given information:

The number $a_n$ of balls per layer is given by $a_n=\frac{n\left(n+1\right)}{2}$ where $n=1$ represents the top layer.

Explanation:

Consider the rule $a_n=\frac{n\left(n+1\right)}{2}$ ,

Substitute the value of n as 5 in the given rule,

  $\begin{array}{c}{a}_n=\frac{n\left(n+1\right)}{2}\\ =\frac{5\left(5+1\right)}{2}\\ =\frac{30}{2}\\ =15\end{array}$

Therefore, the number of balls in the fifth layer is 15.

b.

To state: The total number of balls in the sack consisting five layers.

The resultant balls are 35.

Given information:

The number $a_n$ of balls per layer is given by $a_n=\frac{n\left(n+1\right)}{2}$ where $n=1$ represents the top layer.

Explanation:

To calculate the total number of balls find the sum of ${\displaystyle \sum _{n=1}^5\frac{n\left(n+1\right)}{2}}$ .

  $\begin{array}{c}{\displaystyle \sum _{n=1}^5\frac{n\left(n+1\right)}{2}}=\frac{1\left(1+1\right)}{2}+\frac{2\left(2+1\right)}{2}+\frac{3\left(3+1\right)}{2}+\frac{4\left(4+1\right)}{2}+\frac{5\left(5+1\right)}{2}\\ =\frac{1\cdot 2}{2}+\frac{2\cdot 3}{2}+\frac{3\cdot 4}{2}+\frac{4\cdot 5}{2}+\frac{5\cdot 6}{2}\\ =1+3+6+10+15\\ =35\end{array}$

Therefore, the total number of balls is 35.

c.

To state: The difference between the numbers of balls in a layer of a triangular pyramid with the number of balls in the same layer of a square pyramid.

The resultant difference is $n^2-\frac{n\left(n+1\right)}{2}$ .

Given information:

The given statement says to compare the number of balls in a layer of a triangular pyramid with the number of balls in the same layer of a square pyramid.

Explanation:

The number $a_n$ of balls per layer in a square pyramid is given by $a_n=n^2$ .

Substitute the value of n as $1,2,3,4,5$ in the given rule,

  $\begin{array}{c}{a}_1=1^2,a_1=1\\ a_2=2^2,a_2=4\\ a_3=3^2,a_3=9\\ a_4=4^2,a_4=16\\ a_5=5^2,a_5=25\end{array}$

It can be seen that except first term every other term of the square pyramid have more number of balls than the triangular pyramid.

The difference in number of balls is:

  $n^2-\frac{n\left(n+1\right)}{2}$