Let P(n):11.4+14.7+17.10+....+1(3n−2)(3n+1)=n(3n+1)
For n=1,LHS=11.4=14
RHS=13+1=14
∴LHS=RHS
∴P(1) is true
Lte us assume P(k) is true for some k∈N
i.e, 11.4+14.7+17.10+....+1(3n−2)(3k−2)=n(3k+1)
=k3k+1
Adding (k+1)th term =1(2k+1)(3k+4)
=k3k+1
Adding (k+1)th term =1(2k+1)(3k+4)
on both sides, we get
11.4+14.7+17.10+....+1(3k−1)(3k+1)+1(3k+1)(3k+4)
=k3k+1+1(3k+1)(3k+4)
=k(3k+4)+1(3k−1)(3k+4)=3k2+4k+1(3k+1)(3k+4)
=k+13(k+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.