Let
P(n)=11.2+12.3+13.4+....+1n(n+1)=nn+1
For n=1,LHS=11.2=12;RHS=11+1=12
∴ LHS=RHS
∴ P(1) is true
Let us assume P(k) is true for some k∈N
i..e, 11.2+12.3+13.4+....+1k(k+1)=kk+1
Adding (k+1)th term =1(k+1)(k+2) on both sides,
we get,
11.2+12.3+....+1k(k+1)+1(k+1)(k+2)
=kk+1+1(k+1)(k+2)
=k(k+2)+1(k+1)(k+2)
=k2+2k+1(k+1)(k+2)
=k+1)2k+1)(k+2)=k+1k+2
=k+1k+1)+1
which is P(k+1)
Thus, P(k)⇒P(k+1)
Hence, by mathematical induction P(n) is true for all n∈N