how do I do proofs by induction?

The general method is: 1)write down what needs to be shown (the claim) 2)check it holds for the lowest value of n required (normally n=1 but check question) 3)write down sentence: 'Suppose when n=m the claim holds' 4)Starting from/using 3), obtain the corresponding claim for n=m+1 (e.g. using algebraic manipulation, methods of integration etc.) 5)end with: 'So if the claim holds for n=m it then holds for n=m+1. Since it holds for n=1, by induction we are done.' Example Prove by induction that 12+36+108+...+4x3ⁿ=6(3ⁿ- 1) Solution: step 1) is just the exact question statement. When n=1, the LHS is 4x3=12 and the RHS is 6(3-1)=12=LHS so the claim is true (this is step 2) done). Now suppose that when n=m the claim holds (this is step 3） done). We have 12+36+108+...+4x3^m+4x3^m+1=(12+36+108+...+4x3^m)+4x3^m+1=6(3^m-1)+4x3^m+1  （by our assumption in step 3))                                                                                                  =2x3^m+1-6+4x3^m+1 (expanding the brackets)                                                                                                  =6x3^m+1-6                                                                                                                                =6(3^m+1-1)           (this is step 4) done as this is what we want) So if the claim holds for n=m it then holds for n=m+1. Since it holds for n=1, by induction we are done. (step 5) done).