Proof by Induction, could someone show how to do this?

Question

Can someone please solve this, and explain how it was solved?

t(1) = 2

t(n) = t(n-1)+3n-1 (n>1)

Prove by induction that t(n) = [n(3n+1)]/2

Answer 1

(a) For t(1)

t(1)= [ 1(3+1)]/2 =2 OK.

(b) for n>1,

substitute t(n) to see if it is OK.

t(n)= ( 3 n^2 + n )/2

  = ( ( 3 n^2 - 6n + 3 ) + (n-1) +6n -3 +1 ) / 2

  = (3(n-1)^2 + (n-1))/2 + 3 n - 1  

  = t(n-1) + 3n -1

(c) both OK, Proved by induction.