Chapter 1.1, Problem 15PPSE

Solution

The expression for the number of tiles or circles in the $n$ th figure.

The expression for the number of tiles or circles in the $n$ th figure is $a_n=4n-1$ .

Given:

  

Formula Used:

The formula of Arithmetic sequence is:

  $a_n=a+\left(n-1\right)d$

Where,

  $a\to$ First term

  $d\to$ Common difference

  $n\to$ Number of terms

  $a_n\to$

  $n$ th term

Calculation:

The number of tiny squares in first figure is $3$ .

The number of tiny squares in second figure is $7$ .

The number of tiny squares in third figure is $11$ .

The number of tiny squares in fourth figure is $15$ .

Therefore, the sequence is:

  $3,7,11,15,\mathrm{...}$

Here,

  $a=3$ and $d=4$

Therefore, the $n$ th term will be:

  $a_n=a+\left(n-1\right)d$

Substitute the values:

  $\begin{array}{c}{a}_n=3+\left(n-1\right)\left(4\right)\\ a_n=3+4n-4\\ a_n=4n-1\end{array}$

Therefore, The expression for the number of tiles or circles in the $n$ th figure is $a_n=4n-1$ .