## Math problem for the mentally curious

For those who enjoy a math problem to finish off a great day on this planet, here is a problem for you. I randomly came up with it while watching the democratic debate last night (must have been bored). It is two problems, the second one follows from the first.

## Problem statement!

Suppose you owe me money from a recent lone. Your unfortunate situation lets me demand money from you in either one of two ways. You get the choice to choose one of them .

### Option 1:

You pay me 1 dollar a day for $$n$$ days.

### Option 2.

You pay me 0 cents today (day 1), 1 cent tomorrow (day 2), $$0+1+2$$ cents on day 3, $$0 + 1 + \dots + i + \dots +(n-1)$$ cents on day n.

### The dilemma:

[1]: At what day does it become not profitable for you to pay me with option two.

[2]: Additionally, create an expression to tell me how much money I will profit if you pick option 2. Hint: Negative profits for days 1 and so on, are possible until the day when statement 1 above is true.

## Problem statement! For problem 2

Suppose you owe me money from a recent lone. Your unfortunate situation lets me demand money from you in either one of two ways. You get the choice to choose one of them .

### Option 1:

You pay me $$x$$ dollar a day for $$n$$ days.

### Option 2.

You pay me 0 cents today (day 1), 1 cent tomorrow (day 2), $$0+1+2$$ cents on day 3, $$0+1+ \dots +i+ \dots +(m-1)$$ cents on day m.

### The dilemma:

Find an equation for x and n that calculates what day it becomes not profitable for you to pay me with option two (in other words, figure out which day $$nx = m$$. Additionally, create an expression to tell me how much money I will profit if you pick option 2. Hint: Negative profits for days 1 and so on, are possible until the day where $$mx = n$$.