# Math induction, how does it work? [duplicate]

I'm reading a book(concrete Mathematics) and it uses a lot the concept of 'Math induction'. Thought I've been reading tutorials and examples about it, I'm still unable to understand it, the last 'basic' example I was reading is(my questions are the ones highlighted):

Prove $$1+2+...+n=\frac{n(n+1)}{2}$$

Then it indicates:

• Step 1: Show is valid for n=1: $$1=\frac{1(1+1)}{2}=\frac{2}{2}=1$$
• Step 2: Inductive step, assume is valid for n=k:

$$1+2+...+k=\frac{k(k+1)}{2}$$ //why do I need to substitute n by k? to me is exactly the same as doing nothing, I could work with n+1 instead

• Step 3: Prove is valid for $$n=k+1$$, and the demo is:

$$1+2+...+\color{red}{(k+1)}=(1+2+...+k)+\color{red}{(k+1)}$$

= $$\frac{k(k+1)}{2}+k+1$$

= $$\frac{k(k+1)+2(k+1)}{2}$$

= $$\frac{(k+1)(k+2)}{2}$$//How is getting this from the previous equation???????

= $$\frac{(k+1)[(k+1) + 1]}{2}$$

Any help is appreciated, this 'basic thing' is driving me nuts.

• I mean, I could answer most of these questions, but I don't honestly think they will really elucidate anything. I would check out a intro to induction on google to help you out Oct 9 '18 at 20:09
• $k(k+1) + 2(k+1) = (k+2)(k+1)$ is just basic factorization. Oct 9 '18 at 20:10
• @RushabhMehta As I mentioned, I've been through many examples (most of them showing exactly the same) and still don't get it Oct 9 '18 at 20:21

The principle of induction is based in the Axioms of Peano, an Italian mathematician. It states that, if a set X is such that $$1 \in X$$ and every successor of X is also an element of X, that is: $$s(X) \in X$$, then this set is the set of natural numbers!
So, you could actually work with $$n$$ and $$n+1$$. It's just a matter of mathematical rigour! :)