Chapter 3, Problem 22PT

To determine

To find: Perimeter of a figure that has 6 pentagons.

Answer to Problem 22PT

Perimeter of figure with 6 pentagons = 20cm

Explanation of Solution

Given information:

Length of each side = 1cm.

Concept:

An Arithmetic sequence adds or subtracts a fixed amount (common difference) to get the next term in the sequence.

We know $n^{th}$ term of Arithmetic progression is

  $a_n=a+\left(n-1\right)d$

Where

  $a$ = First term

  $d$ = Common difference

Calculation:

From given figure of one pentagon

We can say that

Perimeter of one pentagon = 5

From given figure of two pentagons

We can say that

Perimeter of figure of two pentagons = 8

From given figure of three pentagons

We can say that

Perimeter of figure of three pentagons = 11.

This forms a series as $5,8,11,\mathrm{....}$

Common difference = $d=8-5=11-8=3$

As difference of any consecutive terms is 3.

So given sequence is Arithmetic.

We need to find the perimeter of figure with 6 pentagons

It means $n=6$

Therefore,

Substituting value in $n^{th}$ term formula

We get

  $\begin{array}{l}{a}_n=a+\left(n-1\right)d\\ \Rightarrow a_6=5+\left(6-1\right)3\\ \Rightarrow a_6=5+5\left(3\right)\\ \Rightarrow a_6=5+15\\ \Rightarrow a_6=20\end{array}$

Therefore,

Perimeter of figure with 6 pentagons = 20cm