# Application of infinite geometric progression

In this section we will discuss about the application of infinite geometric progression. 1) One side of an equilateral triangle is 18 cm. The mid points of its sides are joined to form another triangle whose mid points, in turn , are joined to form still another triangle. The process is continued indefinitely. Find the sum of the perimeters of all the triangles.
Solution : Let $A_{1}A_{2}A_{3}$ be the first equilateral triangle with each side 18 cm. Let $B_{1},B_{2},B_{3}$ are the mid points on its sides and $C_{1},C_{2},C_{3}$ are further midpoints of triangle.

All sides of triangle = 18 cm ⇒ $A_{1}A_{2} = A_{1}A_{3} =A_{2}A_{3}$ = 18 cm
∴ perimeter of $\Delta A_{1}A_{2}A_{3} = 3 \times$ 18 ----- (1)
∴ $B_{1}B_{2} = \frac{A_{2}A_{3}}{2}$;

$B_{1}B_{2} = \frac{18}{2}$

As $\Delta B_{1}B_{2}B_{3}$ is an equilateral triangle so each side of this triangle is $\frac{18}{2}$
∴ perimeter of $\Delta B_{1}B_{2}B_{3} = 3(\frac{18}{2}$) ----- (2)
Now we will consider inner triangle $C_{1}C_{2}C_{3}$
$C_{1}C_{2} = \frac{B_{2}B_{3}}{2}$
OR
$C_{1}C_{2} = \frac{A_{1}A_{2}}{4}$
∴ perimeter of $\Delta C_{1}C_{2}C_{3} = 3(\frac{18}{4}$) ----- (3)
∴ Sum of all the perimeters of the triangle will be
= $\left \{ 3 \times 18 + 3(\frac{18}{2})+ 3(\frac{18}{4}) + ...+ \infty \right \}$
= 3$\left \{ 18 + \frac{18}{2}+ \frac{18}{4} + ...+ \infty \right \}$

2) Find the rational number whose decimal expansion is $0.3\overline{56}$.
Solution : We have,
$0.3\overline{56}$ = 0.3 + 0.056 + 0.0056 + 0.00056 + 0.000056 + ...+ $\infty$
⇒ $0.3\overline{56}$ = 0.3 +$\left [ \frac{56}{10^{3}} + \frac{56}{10^{5}} + \frac{56}{10^{7}} + ...\infty \right ]$

First term = a = $\frac{56}{10^{3}}$ and common ratio = r = $\frac{1}{10^{2}}$

⇒ $0.3\overline{56} = \frac{\frac{56}{10^{3}}}{1-\frac{1}{10^{2}}}$

⇒ $0.3\overline{56} = \frac{3}{10} + \frac{56}{990}$

= $\frac{353}{990}$

## Practice questions on application of infinite geometric progression

1) A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own. (Ans- 2046)

2) A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.
(Ans- Rs 17000; 295000)

3) Find the sum of the following series up to n terms:
(i) 5 + 55 +555 + … (ii) .6 +. 66 +. 666+…

4) A manufacturer reckons that the value of a machine, which costs him Rs. 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years. (Ans-. Rs 5120)