Chapter 11.7, Problem 32PPS

(a)

To determine

Find the number of squares there are in the third figure

Answer to Problem 32PPS

In the 3^rd figure the number of square is 14

Explanation of Solution

Given:

  

There is a total 5 squares in the second figure. How many squares are there in the third figure?

Concept Used:

1^st figure is having one square

2^nd figure is having 5 squares

3^rd figure should be 14 squares.

Thus, in the 3^rd figure the number of square is 14

(b)

To determine

Write a sequence for the first five figures.

Answer to Problem 32PPS

The sequence for the first five figures: 1, 5, 14, 30, 55

Explanation of Solution

Given:

  

Write a sequence for the first five figures.

Concept Used:

1^st figure is having one square

2^nd figure is having 5 squares

3^rd figure should be 15 squares.

4^th figure should be 30 squares.

5^th figure should be 55 squares.

The sequence for the first five figures: 1, 5, 14, 30, 55

Thus, The sequence for the first five figures: 1, 5, 14, 30, 55

(c)

To determine

Find the number of squares in the 8 x 8 checker board.

Answer to Problem 32PPS

The number of squares in a 8 x 8 checker board is 204.

Explanation of Solution

Given:

  

How many squares are there in a standard 8 x 8 checker board?

Concept Used:

The recursive formula for the sequence: $a_n=\frac{n(n+1)(2n+1)}{6}$

Calculation:

The recursive formula for the sequence: $a_n=\frac{n(n+1)(2n+1)}{6}$

Substituten = 8; the number of squares in 8 x 8 checker board is $a_8=\frac{8(8+1)(2·8+1)}{6}=204$

Thus, the number of squares in a 8 x 8 checker board is 204.

(d)

To determine

Write a formula to represent the number of squares in an n x n grid.

Answer to Problem 32PPS

Formula to represent the number of squares in n x n grid is $\frac{n(n+1)(2n+1)}{6}$

Explanation of Solution

Given:

  

Write a formula to represent the number of squares in an n x n grid.

Concept Used:

Formula to represent the number of squares in n x n grid is $\frac{n(n+1)(2n+1)}{6}$

Thus, Formula to represent the number of squares in n x n grid is $\frac{n(n+1)(2n+1)}{6}$