Before doing lots of working out, I would probably choose Option 3 because even though I only start with 1 pence, eventually I will be getting more than 10 pounds a day so it is a better option than Option 1. Also, by day 4 with Option 3, I would be getting more than 50 pence a day so it is better than Option 2.
Now onto solving the problem. I started off by using a spreadsheet. (This spreadsheet is attached and can be referred to when reading my answer). In the first column, I put Option 1. I started with £10, then using the formula:
above cell + 10
I was able to get how much money I would have per day and how much money I would have by the end of the month. I concluded that I would have £310.
Then I went on to Option 2. For this one, I started off by making a column of the amount of money I would get per day, not adding them up. I did this by using the formula:
above cell + 0.5
For all 31 days.
Then, in the column to the left, I put the totals. I found them out by using the formula:
above cell + cell to the right
Now, I had all the totals per day and by the 31st day, I would have £325.5.
Finally, I inputted Option 3 into the spreadsheet. I used the same method as Option 2 to find out my totals. In the first column, I used the formula:
above cell multiplied by 2
To find how much money was added each day.
As done above, I used the formula:
above cell + cell to the right
To find all the totals.
I found that by the end of the month I would have £21,474,836.47.
This means that Option 3 would be the best option as you would have significantly more money compared to the other options by the end of the month.
To further develop my answer, I decided to answer the some of the questions asked.
The first question was:
In which months would option 1 be better than option 2?
Using my spreadsheet I was able to see that on day 29, the money received in Option 1 and Option 2 became equal (they would both have £290). This means that only in February, Option 1 will be better since it is the only month with 28 days.
The second question was:
If your family stopped your pocket money on day 8, which option would give you the most money?
Again, using my spreadsheet, I saw that on day 8, Option 1 would give you the most money. With Option 1, you would receive £80, but with Options 2 and 3, you would only be receiving £38 and £2.55 respectively, concluding that Option 1 would be the better choice.
The third question was:
On which day of the month does Option 3 become the most fruitful?
Just by looking at the spreadsheet, I can see that on day 13, Option 3 will have £81.91, which is more fruitful than Option 2 which will only have £78. While that is true, on day 13, Option 1 is still more fruitful than Option 3 as Option 1 will have £130. However, Option 3 does become more fruitful than Option 1 on day 14 because Option 3 has £163.83 while Option 1 only has £140.
The last question was:
If you chose option 3, how many days would it be before you became a millionaire?
I went through the Option 3 column in my spreadsheet and saw that on day 27, you would be a millionaire as you would have £1,342,177.27. You wouldn't be a millionaire yet on day 26 as you would only have £671,088.63.
In conclusion, Option 3 is always the best choice as you will have significantly more money in all months, including February.