Here's a tricky one which I don't know how to start so any help would be appreciated. Show that no 4 consecutive binomial coefficients can be in AP and no 3 consecutive binomial coefficients can be in GP or HP
Is there a general method to be followed while solving such problems.
For the first part (AP) try thinking about Pascal's Triangle and the line above the one you are looking at. Can you show that you can't have three successive binomial coefficients the same?
For the other two, working with the binomial coefficients you will find that there is mass cancellation when you try to find the condition that the progression exists.