I'm learning about proofs using induction and our professor want us to always use strong induction when giving proofs. In my understanding, strong induction is used to show the range of numbers you have shown to be true. I want to be able to write a complete and clear proof. I'm not really understanding how to correctly write the inductive hypothesis for this problem: for all natural numbers n, prove 1+2+3,+...+ n = n(n+1)/2.
I already showed the base case is true when n = 1. But for the inductive step I want to know if I did it right or not. Even if its correct, is there anything I can add or remove to make it clearer? This is what I have so far:
If n <= k and n >= 1, assume 1+2+3+...+k = k(k+1)/2 is true for 1 <= n <= k.
Prove 1+2+3+...+ k+(k+1) = (k+1)(k+1+1)/2.
I know how to show the proof after this, I just want to understand how to write the inductive hypothesis. Thanks for any help.