tux 2 years ago Prove by induction: http://i.imgur.com/715S5.png

1. mukushla

Proof: For each positive integer $$n\ge2$$ , let $$S(n)$$ be the statement$1.2+2.3+...+(n-1)n=\frac{n(n-1)(n+1)}{3}$ Basis step: S(2) is the statement $$1.2=\frac{2.1.3}{3}=2$$. Thus $$S(2)$$ is true. Inductive step: We suppose that $$S(k)$$ is true and prove that $$S(k+1)$$ is true. Thus, we assume that $1.2+2.3+...+(k-1)k=\frac{k(k-1)(k+1)}{3}$ and prove that $1.2+2.3+...+k(k+1)=\frac{k(k+1)(k+2)}{3}$ what to do now?

2. mukushla

@tux make sense?

3. tux

You rewritten sum as $\sum_{i=1}^{n-1}i(i+1)=\sum_{i=1}^{n}i(i+1)+(n-1)(n-1+1)$ ? And then substituted n=k in induction step Then n=k+1 you got ((k+1)-1)(k+1) k*(k+1)=$\frac{ k(k+1)(k+2) }{ 3 }$

4. mukushla

this is how we got things done by induction We assume that $$S(k)$$ is true and we want to prove that $$S(k+1)$$ is true.

5. tux

Thank you. Now I can do it alone

6. mukushla

yw :)

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