On option 1, you will accumulate £310 at the end of the month (31 x 10) with an nth term of 10N, N being the number of the day.
On option 2, there is an nth term of £2.50 + £0.50N (these are all added together), this will total £315 at the end of the month.
On option 3, there is an nth term of 2^(N-1), making £21,474,836.47 after 31 days of saving.
Option 1 is better than option 2 on any months with equal to or less than 30 days. If we subtract the final 2 amounts of each (310 - 10) and (315-18) leaves £300 and £297, making option 1 better, and therefore proving this works for February as well.
If your parents stopped your pocket money on day 8, option 1 would give the most. If 8 is N, 10N is £80, option 2 makes £22 and option 3 makes £1.28.
On Day 15, option 3 becomes most fruitful because 2^N-1 makes it 2^14, equating £163.84, and option 1 only makes £150 (10 x 15).
You become a millionaire on day 28, as 2^28-1 is 2^27 and >1,000,000 as it is £1,342,177.28.