Chapter 13.8, Problem 78STP

a.

To determine

Find an infinite geometric series to represents the sum of the perimeters of all the triangles.

Answer to Problem 78STP

The geometric series is ${\displaystyle \sum _{k=0}^{n}39{(1.5)}^n}$ .

Explanation of Solution

Given: It is given in the question that there is an equilateral triangle ABC that have a perimeter $39cm$ . If the midpoints of the sides are connected, a smaller equilateral triangle results and also suppose the process of connecting midpoints of sides and drawing new triangles is continued indefinitely.

Concept Used:

In this, use the concept of infinite geometric series formula ${\displaystyle \sum _{k=0}^{n}a{r}^k}$ .

Calculation:

Since, in this the perimeter is increasing after then after so first term will be $a=35$ .

And it is seen that the perimeter is three halves that and it is forever this, $r=\frac{3}{2}=1.5$ . So, the infinite geometric series is ${\displaystyle \sum _{k=0}^{n}39{(1.5)}^n}$ .

Conclusion:

The series is ${\displaystyle \sum _{k=0}^{n}39{(1.5)}^n}$ .

b.

To determine

Calculate the sum of the perimeters of all the triangles.

Answer to Problem 78STP

The perimeter of the triangles is $\infty$ .

Explanation of Solution

Given:

It is given in the question that there is an equilateral triangle ABC that have a perimeter $39cm$ . If the midpoints of the sides are connected, a smaller equilateral triangle results and also suppose the process of connecting midpoints of sides and drawing new triangles is continued indefinitely.

Concept Used:

In this, use the concept of understanding the question and the series and if the same thing is goes on then it becomes infinite.

Calculation:

Here, the series is

  ${\displaystyle \sum _{k=0}^{n}39{(1.5)}^n}$ .

Since, the sum is growing , it is not converging to a particular point because the number being rasied to a power is greater than one, the sum will grow.

So, the perimeter of all the triangles is infinite $\infty$ .

Conclusion:

The perimeter is $\infty$ .