Chapter 11.2, Problem 67PPS

(a)

To determine

Make drawings to find the next three numbers as tables are added one at a time to the arrangement.

Answer to Problem 67PPS

The next three numbers are: $14,18,22$

Explanation of Solution

Given:

The rectangular tables in a reception hall are often placed end − to − end to form one long table. The diagrams below show the number of people who can sit at each of the table arrangements.

  

Make drawings to find the next three numbers as tables are added one at a time to the arrangement.

Concept Used:

  $a_1=6;a_2=10,d=10-6=4$

Calculation:

The next three terms: $a_3;a_4;a_5$   $\begin{array}{l}{a}_1=6;d=4;a_2=10;\text{find}{a}_3;a_4;a_5\\ a_3=a_2+d=10+4=14\\ a4=a_3+d=14+4=18\\ a_5=a_4+d=18+4=22\end{array}$

Figures

Thus, the next three numbers are: $14,18,22$

(b)

To determine

Write an equation representing the nth number in this pattern.

Answer to Problem 67PPS

  $a_n=4n-2$

Explanation of Solution

Given:

The rectangular tables in a reception hall are often placed end − to − end to form one long table. The diagrams below show the number of people who can sit at each of the table arrangements.

  

Write an equation representing the nth number in this pattern.

Concept Used:

The general nth term of an AP series is $a_n=a_1+(n-1)·d$

Calculation:

  $\begin{array}{l}{a}_1=6;d=4;\text{find}{a}_n\\ a_n=a_1+(n-1)·d\\ a_n=6+(n-1)·4\\ a_n=6+4n-4=4n-2\\ a_n=4n-2\end{array}$

Thus, $a_n=4n-2$

(c)

To determine

Find it is possible to have seating for exactly 100 people with such an arrangement.

Answer to Problem 67PPS

No, it is not possible.

Explanation of Solution

Given:

The rectangular tables in a reception hall are often placed end − to − end to form one long table. The diagrams below show the number of people who can sit at each of the table arrangements.

  

Is it possible to have seating for exactly 100 people with such an arrangement?

Concept Used:

No: there is no whole number n for which 4n + 2 = 100

Thus, no, it is not possible.