Chapter 8.1, Problem 27E

To determine

To Find: Which rule gives the total number of squares in the $nth$ figure of pattern shown.

Answer to Problem 27E

  $(D)a_n=\frac{n(n+1)}{2}$ Squares

Explanation of Solution

Given:

  

Here

In the first pattern there is only one square.

In the second pattern there are three squares having one diagonal.

In the third pattern there are six squares having three diagonals and so on…

Thus the pattern form is

  $1+2+3+4+\mathrm{....}+n$ Squares

Which is equal to

  $\frac{n(n+1)}{2}$ Squares.

Hence, $\frac{n(n+1)}{2}$ Squares.