# Math

In a financial deal, you are promised \$800 the first day and each day after that you will receive 45% of the previous day's amount. When one day's amount drops below \$1, you stop getting paid from that day on. What day is the first day you would receive no payment and what is your total income?

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1. f(day)=800*.45^(day)

Set f(day)=1, solve for day by taking a log on both sides.

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2. instead of .45^(day) you should actually use .45^(day-1) so that the first day is 800

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3. I tried doing this, but I still have no idea how to solve this problem =/

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4. e.g.
day 1 = 800
day 2 = 800(.45)
day 3 = 800(.45)^2
looks like day (n+1) would be 800(.45)^n
..

when is 800(.45)^n = 1
.45^n = 1/800 = .00125
take log of both sides
log(.45^n) = log .00125
n log .45 = log .00125
n = log .00125/log .45 = 8.3

checking:
day 9 = 800(.45)^8 = 1.35
day 10 = 800(.45)^9 = .61 ----> receive no income

so he receives income for 9 days
so we want the sum of the geometric series of 9 terms
with a = 800 and r = .45
sum(9) = 800(1 - .45^9)/(1 - .45)
= 1453.44

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5. I understand it now! Thank you so much!

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6. i still don't get it

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