So, for the first question, "Without doing any working out, which one would you choose, and why?", you may think that option 1 or 2 is the best. £10 a day for a month is good, and, even though option 2 starts with £3, we can imagine that it gets bigger and bigger after a few days, so we can expect a decent total at the end.
However, after a bit of thinking (and no working out yet), you should start thinking about option 3.
Option 3 starts at 1p (or £0.01), and then to 2p, and then to 4p, etc, etc. At first, it may seem like you would have to be stupid to pick this option. However, if you know your powers of 2, you know that they get big VERY quickly. Let's take 2^10 pence. That's 1024p, or £10. So after 10 days, our "money per day" has already increased 1000 times! (10/0.01 = 1000). Now imagine what 2^31 is!
Therefore, after a bit of thinking, but no working out, we have deduced that option 3 is definitely the best one. Now let's do some working out for the 31 day month.
The next question aks for some totals. If you see my spreadsheet, I've worked them out, but I'll give a brief expanation here.
For option 1:
This is simple. We're getting £10 a day for 31 days, so we end up with 10*31 pounds, or £310.
For option 2:
This seems simple, but it is a but annoying to show mathematically.
If you know nth term, you should see that we're going up 0.5 every time, so we start with 0.5n. That gives us 0.5,1,1.5,2,etc. To get from 0,5 to 3 we add 2.5, so we have 0.5n + 2.5.
We have the list: 3, 3.5, 4, 4.5, etc etc. We can calculate the last term to get a sense of how much money we are getting. We do 0.5 * 31, which is 15.5, and add 2.5, to get 18.
So we know our minimum value is 3 and our maximum value is 18. The average of this is 3 + 18 = 21, 21 / 2 = 10.5. Therefore, we are getting a little bit more per day than we did in option 1, and if you see my spreadsheet we are. We end up with £326.
Option 3:
As you saw earlier, we will be ending up with extremely large numbers, so there is no point in trying to average and adding everthing up. We can find out the last number of our sequence of values though.
Be aware that our last number in our list will NOT be 2^31 pence. We are starting at 1 pence, which equals 2^0 pence. Therefore, since we are "one behind" our position in the list (the fifth value will be 2^4 pence), our last value will be 2^30. This is extremely important because if we added another 2^31 pence to our list, we will be essentially tripling our total.
So, 2^30 pence = 1073741824 pence, and if we divide that by 100 (to convert to pounds), we get £10,737,418.24. That is a LOT of pocket money!
If we add every day's pocket money, we get £21,474,836.
We clearly have a best option, so let's go onto the next questions.
The next question asks us about option 1 and 2. We have three different lengths of months: 31, 30 and 28 days. In my spreadsheet I have made three different tables for these. As you can see on the last table, for a 28 day month, option 1 is better than option 2. You can think of it as the average for every number in option 2's list getting lower and lower until it gets below the £10 a day we are getting for option 1. This is when option 1 becomes better.
The next question asks us about which option would be better if day 8 was the last day we recieved pocket money. As you can see, the total for option 3 had become very low. This is becuase powers of 2 only get large when the power is higher than about 8. Therefore, we do not get crazily high values, and we don't become a millionaire.
Again, option 1 is easy to calculate, it's just £10 * 8, which is £80. Just like option 3, option 2 does not get us much money. This is becuase the average money per day is low becuase we have not got far in our sequence. As you can see on my spreadsheet, our maximum per day is only £6.50. Therefore, we are going to get considerably less for option 2 than we would for option 1.
The penultimate question asks us on which day of the month is option 3 the most fruitful. Well, our money per day gets larger over time. Therefore, our money per day will be highest on the last day of the month, or the 31st day.
The last question asks us how many days until we are a millionaire. Remember, my spreadsheet doesn't show running totals, so we will have to do some trial and error.
After 26 days, we have a total of £671,088.63. Notice how the total of every day's pocket money before day 27 added up is one pence off the money we get on day 27. This is because you can think about powers of two as timesing 1 by 2 many, many times (i.e. 2*2*2*2*2*2*2...)
Let's just go through some running totals. On the second day, we get 2 pence. This is 1 pence off the total for all the days before it. There is only one day before day 2, which is day 1, where we got 1 pence. Therefore, our total is 1 pence off the next value. Then go on to day 3. We get 4 pence on day 3, which is one pence off the total for the previous days. Our total up to this point is 1 pence + 2 pence, which is 3 pence. Notice how we have just come up with a more efficient, or better, way of working out when we will become a millionaire. SI=ince the next number in the list is very very close to our current total, we can just call them the same. If the next number in the list is over a million, out current total is over a million and we are a millionaire! Huzzah!
Now let's look at day 27. We can use our new technique for this. So we look at the next number, or the money we will get on day 28, which is £1,342,177.28. Boom! Our current total is over £1,000,000, so we are a millionaire!
I don't think our parents have any money left though.