SOLUTION: I stuck on this problem and have been for a while. I hope you can help me out!! A quiz program gives a prize of $5 for teh first question answered correctly. The contestant's pr

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Question 122541: I stuck on this problem and have been for a while. I hope you can help me out!!
A quiz program gives a prize of $5 for teh first question answered correctly. The contestant's prize winnings are doubled for each question after the first. How many questions would you need to answer to exceed $1,000,000 in total prize money?


Found 2 solutions by Fombitz, stanbon:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
This looks like exponential growth.
The formula for exponential growth is
f%28t%29=k%2Ae%5E%28%28alpha%29+t%29
We just need to determine k and alpha
f%281%29=k%2Ae%5E%28%28alpha%29%29=5
f%282%29=k%2Ae%5E%282%28alpha%29%29=10
e%5E%282%28alpha%29%29%2Fe%5E%28alpha%29=10%2F5
e%5E%28alpha%29=2
%28alpha%29=ln%282%29
%28alpha%29=0.693
f%281%29=k%2Ae%5E%28%28alpha%29%29=5
k%2Ae%5E%280.693%29=5
2k=5
k=2.5
f%28t%29=2.5%2Ae%5E%280.693t%29
So, what is t so that f(t)>1000000
2.5%2Ae%5E%280.693t%29%3E1000000
e%5E%280.693t%29%3E400000
0.693t%3Eln%28400000%29
0.693t%3E12.89922
t%3E18.61
The first time would be the 19th correct question.
For 19 correct answers, you would win
f%2819%29=2.5%2Ae%5E%280.693%2819%29%29
f%2819%29=2.5%2Ae%5E%2813.167%29
f%2819%29=1307059.78
The payout would be $1.307,059.78.

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
A quiz program gives a prize of $5 for the first question answered correctly. The contestant's prize winnings are doubled for each question after the first. How many questions would you need to answer to exceed $1,000,000 in total prize money?
5,10,20,40,80,160....
This is a geometric sequence with r=2 and a=5
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Sum Formula:
S(n) = a[1-r^(n)]/[1-r]
Example:
S(2) = 5[1-2^(2)]/[1-2] = 5-3/-1 = 15
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Your Problem:
1,000,000 = 5[(2^x)-1]/(2-1)
200,000 = (2^x)-1
200,001 = 2^x
Take the log of both sides to get:
x*log2 = log 200,001
x = [log200,001]/[log2]
x = 17.6096
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When you answer 18 correctly you will be above 1 million in earnings.
==============
Cheers,
Stan H.