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?

I am so confused about this problem. Could someone please show me the steps?

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

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

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

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

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

I understand it now! Thank you so much!

i still don't get it

To solve this problem, we need to understand the given conditions and then calculate the total income and the day when the payment drops below $1. Let's break it down into steps:

Step 1: Understand the conditions
- On the first day, you are promised $800.
- Each subsequent day, you will receive 45% of the previous day's amount.
- When the amount for any day drops below $1, you stop receiving payment from that day onward.

Step 2: Calculate the daily payments
To find the daily payments, we will start with the amount promised on the first day and apply the 45% reduction for each subsequent day until the payment drops below $1.

Day 1: $800 (promised amount)
Day 2: (45% of Day 1's amount) = 0.45 * $800 = $360
Day 3: (45% of Day 2's amount) = 0.45 * $360 = $162
Day 4: (45% of Day 3's amount) = 0.45 * $162 = $72.90
Day 5: (45% of Day 4's amount) = 0.45 * $72.90 = $32.805
Day 6: (45% of Day 5's amount) = 0.45 * $32.805 = $14.76225
Day 7: (45% of Day 6's amount) = 0.45 * $14.76225 = $6.6430125
Day 8: (45% of Day 7's amount) = 0.45 * $6.6430125 = $2.989355625
Day 9: (45% of Day 8's amount) = 0.45 * $2.989355625 = $1.34571053125

As you can see, the payment drops below $1 on Day 9. Therefore, you stop receiving payment from Day 9 onward.

Step 3: Calculate the total income
To find the total income, we sum up the payments received up to the day when the payments stop.

Total income = $800 + $360 + $162 + $72.90 + $32.805 + $14.76225 + $6.6430125 + $2.989355625 + $1.34571053125

Calculating this sum gives us the total income.

Please note that the calculations might involve rounding the amounts after each day to the appropriate decimal places if required.