Jack drops a super bouncy ball from the top of a 56 m building on to a concrete surface below. Its first rebound is  42 m, and each subsequent rebound is three quarters the height of the previous one.

(i) How high will it rise on the fifth rebound?        

(ii) How far will it travel in total?       

The first term of an arithmetic series is 5 and the 10th term is 4 times the second. Find the common difference.

Janet works out the sum of n terms of a given arithmetic series. Her answer, which is correct, could be:

$$\begin{gathered} A. S_n=2(2^n-1) \\ B. S_n=9-2n \\ C.S_n=8n-n^2 \\ D.S_n=7\times 2^{n-1} \end{gathered}$$

 Connor buys a new car, which begins to depreciate immediately. The value ($ V)

 of the car after t years is given by

 $$\begin{gathered} V=a{e}^{-kt} \\ Where A is initial value \\ K is cons\tan t of depreciation \\ t is the times in year \end{gathered}$$

If the car is worth $30 000 after 5 years and $18 000 after 10 years, find the following:

The initial value of the car. 1

(i) The depreciation of constant 
(ii) The initial value of the car
(iii) How many whole years will it take before the car's value falls below $1 000?

The first three terms of a GP are 0.1, 0.12, 0.144 

(i) Find the 40th term, correct to 1 decimal place. 

(ii) Calculate the sum of the first 40 terms, correct to 1 decimal place.

$$\begin{gathered} In a lucky dip, there are twelve identical envelopes of which only three contain prizes. \\ \left(i\right) Show that if one were to purchase two envelopes the probability of not getting a prize would be \frac{6}{11} . \\ \left(ii\right) What is the probability of getting at least one prize in this case? \end{gathered}$$

The population of a certain insect is growing exponentially according to $N=400e^{kt}$ where t is the time in weeks after the insects are first counted. At the end of five weeks the insect population has doubled.

(i) Calculate the exact value of k. 

(ii) How many insects will there be after 8 weeks? 

(iii) At what rate is the population increasing after 5 weeks-.

$$\begin{gathered} The sum of first n terms of a series is given by S_n = 3n-n^2. \\ \left(i\right)Find the sum of the first 10 terms. \\ \left(ii\right)Find the tenth term. \end{gathered}$$

$$\begin{gathered} Evaluate \sum _{n=2}^4\frac{n}{n+1} \\ Find the limiting sum of the geometric series 500 + 100 + 20 + 4 + · · ·. \end{gathered}$$

$Find the sum of the arithmetic series 1 + 4 + 7 + · · · + 226.$