Chapter 11, Problem 4MCQ

(a)

To determine

Write an equation representing the nth number in this pattern where n is the number of white squares.

Answer to Problem 4MCQ

The equation: $a_n=2n+2$

Explanation of Solution

Given:

Write an equation representing the nth number in this pattern where n is the number of white squares.

Concept Used:

Figure 1: White square: 4, Green square: 1

Figure 2: White square: 6, Green square: 2

Figure 3: White square: 8, Green square: 3

Sequence for the White square: $4,6,8,\mathrm{......}$

Calculation:

Sequence for the White square: $4,6,8,\mathrm{......}$

  $a_n=a_1+(n-1)·d$

  $a_1=4;d=6-4=2$

  $a_n=4+(n-1)·2=4+2n-2=2n+2$

  $a_n=2n+2$

Thus, the equation: $a_n=2n+2$

(b)

To determine

Is it possible to have exactly 84 white squares in an arrangement? Explain.

Answer to Problem 4MCQ

Yes, it is possible

Explanation of Solution

Given:

Is it possible to have exactly 84 white squares in an arrangement? Explain.

Concept Used:

Figure 1: White square: 4, Green square: 1

Figure 2: White square: 6, Green square: 2

Figure 3: White square: 8, Green square: 3

Sequence for the White square: $4,6,8,\mathrm{......}$

Calculation:

Sequence for the White square: $4,6,8,\mathrm{......}$

Here: $a_1=4;d=6-4=2$

  $\begin{array}{l}{a}_n=2n+2\text{and}{a}_n=84;\text{Find}n\\ 84=2n+2\\ 84-2=2n+2-2\\ 82=2n\\ \frac{82}{2}=\frac{2n}{2}\Rightarrow n=41\end{array}$

Yes, it is possible to have exactly 84 white squares in 41th arrangement.

Thus, yes, it is possible to have exactly 84 white squares in 41th arrangement.