p(n) =1+4+7+...+(3n-2)=1/2n(3n-1)

P.T by principle of mathematical induction

to prove that ...............(1)

first we will calculate it for n =1. then we will assume that it is true for n = k.

and with the help of assumption, we will try to show that it also holds for n = k+1

therefore p(1) =1 is true.

let it is true for n = k.

therefore

.....................(2)

thus the result also holds for n = k+1.

therefore it is true for all the integer values of n.

hope this helps you.

cheers!!