Before starting to work out the problem, I decided option 3 was probably the best as your amount of daily pocket money grows exponentially as opposed to linearly. This means that the amount it is increasing by gets bigger each day and will eventually surpass option 2 which increases by a flat amount. Due to this property, it means the amount of pocket money I get every day will be larger than option 1 or 2 after a while. I then chose to use an excel spreadsheet to work out how much each option will give me each day and the amount they’ll have given me total.
The first question asks how much pocket money you will have received after 31 days of each option. Using the spreadsheet I checked the total of each option for day 31 to find out you will receive £310 with option 1, £325.50 for option 2 and a massive £21,474,836.47 for option 3! This easily makes option 3 the best followed by option 2 and finally option 1.
After 1 month, option 1 would only be better than option 2 if it was February as after 28 days option 1 will give you £280 and option 2 £273. An exception to this is if it’s a leap year in which case both options will give you £290 pounds in February after 29 days. Any month with 30 or 31 days will cause option 2 to become better.
If your pocket money stopped after 8 days, option 1 would easily be the best giving you £80. Option 2 would be 2nd best with £38 and option 3 would only give you £2.55.
Option 3 becomes the best option on day 14 earning you £163.83, overtaking option 1s £140 and option 2s £87.50. Option 3 will cause you to become a millionaire on day 27, after which you will have amassed £1,342,177.27.
I used an excel spreadsheet to work out all my answers but you could also compare the options in the form of a graph. If you already have the data on excel you can create graphs on that as a different way to compare the data. Alternatively, you could plot the graph yourself from the start putting money on the y axis and days on the x axis. Then, for option 1 you can use y=10x as the equation of the line. For option 2 we need y to equal the sum of the amount of pocket money you get each day up to and including x where x is the number of days. To find out the sum of all the days up to and including x we can use the sum of an arithmetic series rule which is x((a1+ax)/2) where a1 is the first term and ax the xth term. We can find out ax with the arithmetic progression rule which states that ax= a1 + d(x-1). d is the common difference between each term. Combining these 2 rules and substituting in the first term and difference which we know, we get y=x((3+(3+0.5(x-1)))/2) as the equation of our line. For option 3 we also need to find the sum of all the days up to and including x. The sum of the numbers in a sequence where to find the next term you multiply the previous one by 2 is always going to be 2(a1*2^x-1)-a1 so in our case the equation of the line is y=2(0.01*2^x-1)-0.01 for option 3. Plot these onto a graph(an online programme will help) and compare each line for the number of days asked in the question to see which one is the best.