Q20. - Algebra 1

Question

Write a formula that can be used to find the perimeter of a figure containing $n$ squares.

Step-by-Step Solution

Answer

A formula that can be used to find the perimeter of a figure containing $n$ squares is $P=2n+2$.

Step 1. Find the first term of the arithmetic sequence.

It can be observed that the perimeters of a figure containing 1, 2 and 3 squares are 4, 6 and 8 respectively.

It can be observed that the sequence formed is: 4,6,8,…

Step2. Find the common difference.

The sequence 4,6,8,… is an arithmetic sequence because there is a constant difference of 2 between each term of the sequence. Therefore, the common difference of the sequence is 2.

From the arithmetic sequence 4,6,8,…, it can be noticed that the first term of the sequence is 4.

Therefore, the first term and the common difference of the arithmetic sequence 4,6,8,… are 4 and 2 respectively.

Step 3. Find a formula that can be used to find the perimeter of a figure containing n squares .

A formula that can be used to find the perimeter of a figure containing squares is equal to the $n\mathrm{th}$ term of the arithmetic sequence 4,6,8,….

The $n\mathrm{th}$ term $\left(a_n\right)$ of the arithmetic sequence is given by:

$a_n=a_1+\left(n-1\right)d$, where $a_1$ is the first term and $d$ is a common difference.

Therefore, the $n\text{th}$ term $\left(a_n\right)$ of the arithmetic sequence 4,6,8,… having $a_1=4$ and $d=2$ is:  

$a_n=a_1+\left(n-1\right)d$

$$\begin{gathered} =4+\left(n-1\right)\left(2\right) \\ =4+2n-2 \\ =2n+4-2 \\ =2n+2 \end{gathered}$$

Therefore, a formula that can be used to find the perimeter of a figure containing $n$ squares is $P=2n+2$.