The first prize in a lottery is $250 000. Each winner chosen after the first is paid 20% as much as the winner before them.
b) Determine an explicit formula for the general term. (2 marks)
c) Determine a formula for the sum of n terms of the geometric sequence. (2 marks)
d) If 5 winning numbers are chosen,
i) how much will the last person chosen be paid? (1 mark)
ii) how much will have been paid out in the lottery?(1 mark)
Clearly, the nth winner gets
250000 * 0.20^(n-1)
This is just a geometric series with
a = 250000
r = 1/5
So, just apply your seies formulas
To determine the explicit formula for the general term of the sequence, we need to understand the pattern and relationship among the terms.
First, let's identify the pattern: each winner chosen after the first is paid 20% as much as the winner before them. This means the terms of the sequence are decreasing and connected through a common ratio of 0.2.
a) Determine an explicit formula for the general term:
The first term is $250,000, and each subsequent term is 20% (or 0.2) of the previous term. Hence, the explicit formula for the general term can be expressed as:
T(n) = a * r^(n-1)
where:
T(n) represents the nth term of the sequence,
a represents the first term ($250,000),
r represents the common ratio (0.2),
and (n-1) represents the position of the term in the sequence.
So the explicit formula for the general term is:
T(n) = $250,000 * 0.2^(n-1).
b) Determine a formula for the sum of n terms of the geometric sequence:
To find the formula for the sum of n terms of a geometric sequence, we can apply the following equation:
Sn = a * (1 - r^n) / (1 - r)
where:
Sn represents the sum of the first n terms,
a represents the first term ($250,000),
r represents the common ratio (0.2),
and n represents the number of terms.
So the formula for the sum of n terms of the geometric sequence is:
Sn = $250,000 * (1 - 0.2^n) / (1 - 0.2).
c) If 5 winning numbers are chosen:
i) To find out how much the last person chosen will be paid, we need to substitute the value of n into the explicit formula for the general term.
T(5) = $250,000 * 0.2^(5-1) = $250,000 * 0.2^4 = $250,000 * 0.0016 = $400.
Therefore, the last person chosen will be paid $400.
ii) To determine how much will have been paid out in the lottery, we need to calculate the sum of the first five terms using the formula for the sum of n terms.
Sn = $250,000 * (1 - 0.2^5) / (1 - 0.2) = $250,000 * (1 - 0.0016) / 0.8 = $250,000 * 0.9984 / 0.8 = $311,040.
Therefore, the total amount paid out in the lottery will be $311,040.