Arithmetic/Geometric Series Applications

A question is this type if and only if it involves applying arithmetic or geometric series formulas to real-world contexts (prize money, bouncing balls, donations) with explicit model comparisons or sum-to-infinity calculations.

2 questions · Standard +0.8

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CAIE P1 2011 June Q8
7 marks Standard +0.8
8 A television quiz show takes place every day. On day 1 the prize money is \(\\) 1000$. If this is not won the prize money is increased for day 2 . The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money. Model 1: Increase the prize money by \(\\) 1000$ each day.
Model 2: Increase the prize money by \(10 \%\) each day.
On each day that the prize money is not won the television company makes a donation to charity. The amount donated is \(5 \%\) of the value of the prize on that day. After 40 days the prize money has still not been won. Calculate the total amount donated to charity
  1. if Model 1 is used,
  2. if Model 2 is used.
CAIE P1 2015 November Q6
8 marks Standard +0.8
6 A ball is such that when it is dropped from a height of 1 metre it bounces vertically from the ground to a height of 0.96 metres. It continues to bounce on the ground and each time the height the ball reaches is reduced. Two different models, \(A\) and \(B\), describe this. Model A: The height reached is reduced by 0.04 metres each time the ball bounces.
Model B: The height reached is reduced by \(4 \%\) each time the ball bounces.
  1. Find the total distance travelled vertically (up and down) by the ball from the 1st time it hits the ground until it hits the ground for the 21st time,
    (a) using model \(A\),
    (b) using model \(B\).
  2. Show that, under model \(B\), even if there is no limit to the number of times the ball bounces, the total vertical distance travelled after the first time it hits the ground cannot exceed 48 metres.