Geometric Sequences & Series

[Maximum mark: 6]

A tennis ball bounces on the ground $n$ times. The heights of the bounces, $h_1, h_2, h_3, \dots,h_n,$ form a geometric sequence. The height that the ball bounces the first time, $h_1$, is $80$ cm, and the second time, $h_2$, is $60$ cm.

1. Find the value of the common ratio for the sequence. [2]
   

2. Find the height that the ball bounces the tenth time, $h_{10}$. [2]
   

3. Find the total distance travelled by the ball during the first six bounces (up and down). Give your answer correct to $2$ decimal places. [2]
   

[Maximum mark: 6]

A $3$D printer builds a set of $49$ Ei$\text{f}$fel Tower Replicas in different sizes. The height of the largest tower in this set is $64$ cm. The heights of successive smaller towers are $95$ % of the preceding larger tower, as shown in the diagram below.

1. Find the height of the smallest tower in this set. [3]
   

2. Find the total height if all $49$ towers were placed one on top of another. [3]
   

[Maximum mark: 6]

Peter is playing on a swing during a school lunch break. The height of the first swing was $2$ m and every subsequent swing was $84$ % of the previous one. Peter's friend, Ronald, gives him a push whenever the height falls below $1$ m.

1. Find the height of the third swing. [2]
   

2. Find the number of swings before Ronald gives Peter a push. [2]
   

3. Calculate the total height of swings if Peter is left to swing until coming
   to rest. [2]
   

[Maximum mark: 6]

The sides of a square are $8$ cm long. A new square is formed by joining the midpoints of the adjacent sides and two of the resulting triangles are shaded as shown. This process is repeated $5$ more times to form the right hand diagram below.

1. Find the total area of the shaded region in the right hand diagram above. [3]
   

2. Find the total area of the shaded region if the process is repeated indefinitely.[3]
   

[Maximum mark: 6]

A bouncy ball is dropped out of a second story classroom window, $5$ m off the ground. Every time the ball hits the ground it bounces $89$ % of its previous height.

1. Find the height the ball reaches after the $11$th bounce. [3]
   

2. Find the total distance travelled by the ball until it comes to rest. [3]