Learning Resources

Proof by Induction

Step 3

It is important for you to realize that  means j takes on all the integer values from 1 to (k + 1) and k is the integer that comes just before (k + 1). Hence, when expanding the sigma notation result of , include the second last term, 2k -1, in the series before writing the last term of 2(k + 1) - 1. This will allow you to substitute the expression from step 2 obtained by the expansion of .