Question

Prove by mathematical induction that p sub n is true for all n E N:
2 + 4 + 6 + ... + 2n = n(n+1)

Answer 1

\[p(n)=2+4+6+...+2n=n(n+1)\\first \ step \ is \ \check \ for\ n=1\\p(1)=2=1(1+1)\\suppose \ for \ n=k \\ p(k)=2+4+6+...+2k=k(k+1)\\prove \for \ n=k+1\\p(k+1)=2+4+6+...+2k+2(k+1)={2+4+6+...+2k}+2(k+1)=\\{k(k+1)}+2(k+1)=\\(k+1)(k+2)\\comlete\]