Question

Prove by induction:
http://i.imgur.com/715S5.png

Answer 1

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?

Answer 2

@tux
make sense?

Answer 3

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)=\[\frac{ k(k+1)(k+2) }{ 3 }\]

Answer 4

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.

Answer 5

Thank you. Now I can do it alone

Answer 6

yw :)