Chapter 1.1, Problem 36PPSE

Solution

a.

To draw the next two figures of the pattern given in the problem.

The next two figures of the pattern:

Given the pattern:

  

Calculation:

Observe the pattern:

The pattern is constructed of identical squares arranged symmetrically. Each figure of the pattern is constructed from the previous one by adding an extended copy of the last row to the bottom of it extended by two squares on either ends of it.

Thus, the fourth figure will have all the squares of the third one and one more row added to the bottom of it containing $5+2=7$

squares as drawn below:

  

Likewise, the fifth figure of the pattern will have all the squares of the fourth one and one more row added to the bottom of it containing $7+2=9$

squares as:

  

Thus, the fourth and fifth figures of the pattern:

  

Conclusion:

The next two figures of the pattern:

  

b.

To copy and complete the table given in the problem.

The completed table:

Figure (Input)	Process Column	Number of squares (Output)
$1$	$1$	$1$
$2$	$2$	$4$
$3$	$3$	$9$
$4$	$4$	$16$
$5$	$5$	$25$

Given the incomplete table:

Figure (Input)	Process Column	Number of squares (Output)
$1$	$-$	$-$
$2$	$-$	$-$
$3$	$-$	$-$
$4$	$-$	$-$
$5$	$-$	$-$

Calculation:

Observe that the first figure of the pattern comes in the first column and it has only one square.

Thus, the process column of the input $1$ is $1$ and the output is also $1$ .

Now, note that that the second figure of the pattern comes in the second column and it has only four squares.

Thus, the process column of the input $2$ is $2$ for and its output is $4$ .

Proceed in a similar way and find all the process column values and output values corresponding to each other inputs: $3,4,\text{and }5$ .

Incorporate all the findings into the table and complete it:

Figure (Input)	Process Column	Number of squares (Output)
$1$	$1$	$1$
$2$	$2$	$4$
$3$	$3$	$9$
$4$	$4$	$16$
$5$	$5$	$25$

Conclusion:

The completed table:

Figure (Input)	Process Column	Number of squares (Output)
$1$	$1$	$1$
$2$	$2$	$4$
$3$	$3$	$9$
$4$	$4$	$16$
$5$	$5$	$25$

c.

To determine the number of squares in the $n^{\text{th}}$ figure.

It is observed from the table of inputs and outputs of the pattern that the output is $n^2$ when the input is $n$ .

Then, it is concluded that the $n^{\text{th}}$ figure in the pattern has $n^2$ squares.

Calculation:

Observe the table found in the previous part:

The output of $1$ is $1=1^2$

The output of $2$ is $4=2^2$ .

Analysing the table in a similar way, it may be observed that, the output of $n$ is $n^2$ .

Conclusion:

It is observed from the table of inputs and outputs of the pattern that the output of the pattern is $n^2$ when the input is $n$ .

Then, it is concluded that the $n^{\text{th}}$ figure in the pattern has $n^2$ squares.