1.1) compound interest is when interest is earned on top of interest already earned. For example, if I have an account with £100 in at 2% interest per year, after a year I'll have £102 in it. The second year's interest is then 2% of £102, and the third year's will be 2% of that and so on. The longer money is left untouched, the more there will be in each interest payment.
1.2) Following on from this, if you want to know the amount you'll have after a number of years, you can simply chain the multiplications together. For the example above, it would be 1021.021.02 to get the amount after two years. As you've probably spotted, you can simplify the 1.02s to 1.022. This leads into the formula given in part 2 which is the basic formula for compound interest calculation.
4.1) I can sketch this if you like, let me know if you need it.
4.2) The number of outer squares for stages 1-10 is 2,4,8,16,32,64,128,256,512,1024
4.3) The number of outer squares doubles each time and starts at 2, so it's simply the series of powers of 2, which can be written as 2n