91影视

An equilateral triangle is divided into 4 smaller triangles by connecting the midpoints of opposite sides. After that, the other triangles are divided into 4 halves, leaving the center triangle empty. The cycle can continue indefinitely.

Make a diagram for each division to determine the infinite series.

The total of the terms in an infinite geometric series of the form,

$a,ar,a{r}^2,....,\text{to}鈥�鈭�$ 鈥︹�� (1)

Is provided by the expression,

$S=\frac{a}{1-r}$ 鈥︹�� (2)

Consider an equilateral triangle ABC. The midpoints D,E and F of the sides AB, BC, and CA are joined to create 4 equilateral triangles as shown in figure 1 below.

Figure 1

The fourth triangle from the center is left blank, while the others are divided into smaller triangles.

Assume that the triangle ABC has an area of one. Because each triangle is divided into four equal triangles, it has an area of $\left(\frac{1}{4}\right)$each.

As a result, after the first step, the area of the triangle left unfilled is $\left(\frac{1}{4}\right)$.

In the same way, divide triangles 1, 2, and 3. Figure 2 illustrates this.

Figure 2

There are four triangles in each of the triangles 1, 2, and 3. The triangles in the center (shown in brown) are left blank before the next division. The three triangles, each with a total area of $\left(\frac{3}{4}\right)$, are divided into four smaller triangles, each with a total area of $\left(\frac{1}{4}\right)$that is left blank.

As a result, the total area left blank after the second step is $\left(\frac{1}{4}\right)+\left(\frac{3}{4}\right)\left(\frac{1}{4}\right)$. The process is repeated, as seen in Figure 3.

Figure 3

The brown triangles are left empty, while the other nine triangles are each divided into four triangles. The green triangles are left empty for the next stage.

Hence the total area left blank after the third step is provided by,

$\left(\frac{1}{4}\right)+\left(\frac{3}{4}\right)\left(\frac{1}{4}\right)+\left(\frac{3}{4}\right)\left(\frac{3}{4}\right)\left(\frac{1}{4}\right).$

The area of the triangle left blank after each step generates an infinite series if the process is repeated indefinitely, as shown:

$\left(\frac{1}{4}\right),\left(\frac{1}{4}\right)\left(\frac{3}{4}\right),\left(\frac{1}{4}\right){\left(\frac{3}{4}\right)}^2,....\text{to}鈥�鈭�$ 鈥︹�� (3)

Compare the series shown in (3) to that in (1).

Since, the first term is $a=\frac{1}{4}$and $r=\frac{3}{4}$.

Put the values of a and r in equation (2),

$\begin{array}{rcl}S& =& \frac{a}{1-r}\\ & =& \frac{\frac{1}{4}}{1-\frac{3}{4}}\\ & =& 1\\ & & \end{array}$