I am retaking a discrete math class and my new teacher is teaching mathematical induction differently than my old teacher. I really like the way I first learned it because I think it is more formal and I understand it better. But my current teacher says it is wrong, but I know it’s not. I’m fuming and desperately want to prove him wrong. Can you please help me prove that this method is correct?
Example:
Use mathematical induction to prove $\sum_{k=1}^{n} (2k-1) = n^2$
Base Case: $$ \sum_{k=1}^{1} (2k -1) = 1^2 $$ $$ 2(1)-1 = 1$$ $$ 1=1 $$
Inductive hypothesis: Assume $$\sum_{k=1}^{n} (2k-1) = n^2$$
Inductive step: Show the equation holds true for the $n+1$ case.
$$\sum_{k=1}^{n+1} (2k-1) \stackrel?= (n+1)^2$$ $$2(n+1)-1 + \sum_{k=1}^{n} (2k-1) \stackrel?= (n+1)^2$$ $$(2n+1) + \sum_{k=1}^{n} 2k -1 \stackrel?= n^2 + (2n+1)$$ $$\sum_{k=1}^{n} 2k-1 = n^2$$
Therefore if the $n+1$ case holds equivalence from the $n$ case (we can let n=1 because we know that is true), then it must hold true for the $n+2$ case, the $n+3$ case, and so on. But my professor says it’s wrong because I’m changing both sides at once. Can someone help me prove this process?
Instead of an equals sign with a question mark showing what you want to prove, you should start with one side of the equation and produce the other. Given your assumption, you can then write $$\sum_{k=1}^{n+1}2k-1=\left(\sum_{k=1}^{n}2k-1\right)+2n+1\\ =n^2+2n+1\\ =(n+1)^2$$ You should not go from what you hope to prove to what you know, because some steps may not be reversible. Here they are, so you can just do the steps in reverse order. You should either start with an equation you know to be true, like your last, and finish with what you want to prove, or start with an expression and compute the other side of the equation. I did the second of these above.