# Discrete mathematics - understanding proof by induction

So I have an example of a proof that my teacher used induction to solve, but I'm having trouble understanding the inductive step in the second slide. So I get the part where they substitute k+1 in the formula, but I don't understand where the 1+2+...+ 𝒌 +(𝒌 +1) came from and how that managed to turn into the equation in my 2nd underline?

Like the k + (k+1) portion doesn't look like it matches the original formula, and I'm confused because it looks like they substituted n with k instead of k + 1 like it was stated at the beginning of the second slide? I also don't really understand recursion, so I'd appreciate if someone explained the steps behind that transition to me as well.

When they write $$1+2+\ldots + (k+1)$$, they mean summing up every positive integer up to $$k+1$$. The integer before $$k+1$$ is $$k$$.
Hence $$1+2+\ldots + (k+1) =(1+2+\ldots + k)+ (k+1)\tag{1}$$
We already know that $$1+2+\ldots + k = \frac{k(k+1)}{2}$$ by the induction hypothesis, substitute this into $$(1)$$ and we have
$$(1+2+\ldots + k)+ (k+1)= \frac{k(k+1)}{2}+(k+1)$$
My understanding is that- In your first underline they are considering the sum $$1+2+3..$$ till $$(k +1)$$ . This series can be written as $$1+2+3+...+(k+1)$$ = $$1+2+3+...+k+(k+1)$$.
In the second underline, they substituted $$1+2+3+.. +k$$ as $$\frac{k(k+1)}{2}$$, which was derived from the "Inductive Hypothesis".