I don't understand how proof by mathematical induction works, can you help?

This took me some time to get my head around so I know what you mean. I always like to think of proof by induction like dominoes. Once you've pushed the first domino in line of dominos, you know it's going to carry on until the end because every one 'knocks down' the one infront. This is how induction works. So you first show the case for n=0. This is you essentially 'knocking down' the first domino. After that you assume it's true for n = k and show that when it's true for n = k it's also true for n = k+1 i.e. when one domino is knocked down, we know it will knock down the one infront. Once this is done we know it's true for the first case. Our n = k workings show that it is then true for the second case, which then shows it's true for the third case and so on and then it is proven. 

Does this make sense? Do you want to run through a full example?