91Ó°ÊÓ

In geometry, perimeter can be defined as the path or the boundary that surrounds a shape. It can also be defined as the length of the outline of a shape.

Perimeter of the figure with 1 pentagon is,

$$\begin{gathered} P_1=5×1 \\ =5\text{ â¶Ä‰c³¾} \end{gathered}$$

Perimeter of the figure with 2 pentagons is,

$$\begin{gathered} P_2=\left(4×1\right)+\left(4×1\right) \\ =4+4 \\ =8\text{ c³¾} \end{gathered}$$

Perimeter of the figure with 3 pentagons is,

$$\begin{gathered} P_3=\left(4×1\right)+\left(4×1\right)+\left(3×1\right) \\ =4+4+3 \\ =11\text{ c³¾} \end{gathered}$$

Therefore, the perimeters are 5 cm, 8 cm and 11 cm respectively.

First find the difference between the perimeters of corresponding figures.

$\begin{array}{l}d=P_2−P_1=8−5=3\\ d=P_3−P_2=11−8=3\end{array}$

Notice that the common difference is constant. Therefore the sequence of perimeters are in Arithmetic progression.

The perimeter of n polygon$\left(P_n\right)$ is given by

$P_n=P_1+\left(n−1\right)d$

Where d is the common difference.

Here$P_{1.}=5$and$d=3$

$$\begin{gathered} P_{n.}=5+\left(n−1\right)3 \\ =5+3n−3 \\ {P}_{n.}=3n+2\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}…\left(1\right) \end{gathered}$$

In order to calculate the perimeter of 6 pentagons substitute $n=6$in equation

$$\begin{gathered} P_6=3\left(6\right)+2 \\ \text{ }=18+2 \\ {P}_6=20 \end{gathered}$$

Therefore, the perimeter of the figure that has 6 pentagons is 20 cm.

Option His correct.