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.
a) Determine t1 and r for the geometric sequence that represents this situation.
b) Determine an explicit formula for the general term.
c) Determine a formula for the sum of n terms of the geometric sequence.
d) If 5 winning numbers are chosen,
i) how much will the last person chosen be paid?
t1 = 250000
r = 0.20
Now just plug those into the various formulas you have learned.
a) To determine the values of t1 and r for the geometric sequence that represents this situation, we need to understand the pattern in the prize amounts.
Given that the first prize is $250,000, the subsequent winners are paid 20% as much as the winner before them. This means that each successive prize is obtained by multiplying the previous prize by 0.2.
Let's break it down step by step:
t1 = $250,000 (the first prize amount)
r = 0.2 (the common ratio)
b) To find an explicit formula for the general term of the geometric sequence, we can use the formula:
tn = t1 * r^(n-1)
where tn represents the nth term of the sequence.
Substituting the values we found in part (a), the explicit formula for the general term is:
tn = $250,000 * 0.2^(n-1)
c) To determine a formula for the sum of n terms of the geometric sequence, we can use the formula:
Sn = t1 * (1 - r^n) / (1 - r)
where Sn represents the sum of the first n terms of the sequence.
Substituting the values we found earlier, the formula for the sum of n terms is:
Sn = $250,000 * (1 - 0.2^n) / (1 - 0.2)
d) If 5 winning numbers are chosen, we need to find the amount the last person chosen will be paid using the explicit formula found in part (b).
Substituting n = 5 into the formula:
t5 = $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.