2
Do all questions. Hints are at the end of the page. Only refer if really stuck!
Hints:
1.
Submit to Math Rep by 9 Feb, Monday (Arts class) and 12 Feb, Thursday (Science class).
1. A convergent geometric series has first term a and common ratio r . Given that twice the sum of the first and fourth term is equal to three times the sum of the second and third terms, find the value of r . Hence state the sum to infinity of the series. [5]
2. Find the sum of the arithmetic progression 1, 4, 7, 10, 13, 16, …, 100. [3]
Every third term of the above progression is removed, i.e. 7, 16, …, etc. Find the sum of the remaining terms. [3]
3. When a table tennis ball is dropped vertically onto a table, the time interval between any particular bounce and the next bounce is 90% of the time interval between the particular bounce and the preceding bounce. The interval between the first and second bounces is 2 seconds. Given that the interval between the nth bounce and ( n +1) bounce is the first such interval less than 0.02 seconds, find n .
Find the total time taken from the first bounce until the ball stops bouncing.
Is the time found reasonable? Give a reason.
[2]
[2]
[1]
[Do attempt the last part too – something which you may not be sure but give a shot.]
2.
3.
Think whether you can cancel the common factor. There are many ways to solve an equations. Convergent is a very important word here, please check what it means in the context of a GP.
Those you removed formed a progression! You need to know how many terms you are summing at each stage.
It is easier than what you thought for the first two parts.

  n

1 
0.02
   n

1 
0.01
Take ln on both sides (does not change the inequality sign as it is an increasing function)
( n
    ln(0.01) n 1
   n

44.7
n

45
Time for it to stop bouncing = sum to infinity = 2 / (1

0.9) = 20 seconds
In reality, the ball stops bouncing in less than 20s. The model given in the question is unrealistic. Firstly, the model assumes the ball bounces forever as time between the two consecutive bounces will never be 0. Secondly, the ratio of the consecutive times is a constant which is unlikely as the energy loss unlikely to be constant throughout.
Since both assumptions fail, the model is inappropriate hence the time estimated using sum of infinity is unreasonable.