Chapter 8, Problem 13CT

a.

To determine

Describe the meaning of $n$ and $a_n$ in the given pattern.

Answer to Problem 13CT

  $n$ represents the number of blue pattern in each quilt and $a_n$ represents the blue squares.

Explanation of Solution

Given:

The given pattern is

  

Calculation:

In the given pattern, $n$ represents the number of blue pattern in each quilt and $a_n$ represents the blue squares.

b.

To determine

Draw a table for the values of $n$ and $a_n$ .

Explanation of Solution

Given:

The given pattern is

  

The given values of $n=1,2,3,4,5,6,7,\text{and}8$ .

Calculation:

Find the values of $a_n$ for the given values of $n=1,2,3,4,5,6,7,\text{and}8$ .

From the pattern,

  $\begin{array}{l}0+1=1\\ n=1,a_n=1\\ 1+1=2\\ n=2,a_n=2\\ 2+3=5\\ n=3,a_n=5\\ 5+3=8\\ n=4,a_n=8\end{array}$

Similarly

  $\begin{array}{l}8+5=13\\ n=5,a_n=13\\ 13+5=18\\ n=6,a_n=18\\ 18+7=25\\ n=7,a_n=25\\ 25+7=32\\ n=7,a_n=32\end{array}$

Hence the table for the values of $a_n$ with respect to $n$ is given below.

$n$	$a_n$
$\begin{array}{l}1\\ 2\\ 3\\ 4\\ 5\\ 6\\ 7\\ 8\end{array}$	$\begin{array}{l}1\\ 2\\ 5\\ 8\\ 13\\ 18\\ 25\\ 32\end{array}$

c.

To determine

Find the values of $a_n$ for the values of $n$ using given rule and compare with the result $b$ .

Answer to Problem 13CT

The given rule represents the $n\text{th}$ number of blue squares.

Explanation of Solution

Given:

The given pattern is

  

The given rule is $a_n=\frac{n^2}{2}+\frac{1}{4}\left[1-{\left(-1\right)}^n\right]$ and the values of $n=1,2,3,4,5,6,7,\text{and}8$ .

Calculation:

Find the values of $a_n$ for the given values of $n=1,2,3,4,5,6,7,\text{and}8$ .

From the pattern,

  $\begin{array}{l}0+1=1\\ n=1,a_n=1\\ 1+1=2\\ n=2,a_n=2\\ 2+3=5\\ n=3,a_n=5\\ 5+3=8\\ n=4,a_n=8\end{array}$

Similarly

  $\begin{array}{l}8+5=13\\ n=5,a_n=13\\ 13+5=18\\ n=6,a_n=18\\ 18+7=25\\ n=7,a_n=25\\ 25+7=32\\ n=7,a_n=32\end{array}$

Hence the table for the values of $a_n$ with respect to $n$ is given below.

$n$	$a_n$
$\begin{array}{l}1\\ 2\\ 3\\ 4\\ 5\\ 6\\ 7\\ 8\end{array}$	$\begin{array}{l}1\\ 2\\ 5\\ 8\\ 13\\ 18\\ 25\\ 32\end{array}$

Find the values of $a_n$ using rule $a_n=\frac{n^2}{2}+\frac{1}{4}\left[1-{\left(-1\right)}^n\right]$ .

  $\begin{array}{l}n=1,a_1=\frac{1^2}{2}+\frac{1}{4}\left[1-{\left(-1\right)}^1\right]=1\\ n=2,a_2=\frac{2^2}{2}+\frac{1}{4}\left[1-{\left(-1\right)}^2\right]=2\end{array}$

  $\begin{array}{l}n=3,a_3=\frac{3^2}{2}+\frac{1}{4}\left[1-{\left(-1\right)}^3\right]=5\\ n=4,a_4=\frac{4^2}{2}+\frac{1}{4}\left[1-{\left(-1\right)}^4\right]=8\end{array}$

  $\begin{array}{l}n=5,a_5=\frac{5^2}{2}+\frac{1}{4}\left[1-{\left(-1\right)}^5\right]=13\\ n=6,a_6=\frac{6^2}{2}+\frac{1}{4}\left[1-{\left(-1\right)}^6\right]=18\\ n=7,a_7=\frac{7^2}{2}+\frac{1}{4}\left[1-{\left(-1\right)}^7\right]=25\\ n=8,a_8=\frac{8^2}{2}+\frac{1}{4}\left[1-{\left(-1\right)}^8\right]=32\end{array}$

From the above results, the results are same as the part $b$ .

Hence the given rule represents the $n\text{th}$ number of blue squares.