Discrete mathematics - understanding proof by induction

Question

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+...+ +( +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.

Answer 1

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

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

Answer 2

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".