Prove that 1+2+3++n = sum_k=1ⁿ k = (n(n+1))/(2), n 1.

Here are the proofs using the Principle of Mathematical Induction.

1. Prove that $1+2+3+\dots+n = \sum_{k=1}^{n} k = \frac{n(n+1)}{2}$, $\forall n \ge 1$.

Let $P(n)$ be the statement $1+2+3+\dots+n = \frac{n(n+1)}{2}$.

• Base Case (n=1): LHS $= 1$ RHS $= \frac{1(1+1)}{2} = \frac{1(2)}{2} = 1$ Since LHS = RHS, $P(1)$ is true.

• Inductive Hypothesis: Assume $P(m)$ is true for some positive integer $m$. That is, assume $1+2+3+\dots+m = \frac{m(m+1)}{2}$.

• Inductive Step: We need to prove that $P(m+1)$ is true.