give the recursively defined sequence a 1 =1, a 2 =4, and a n =2a n-1 -a n-2 +2, use complete induction to prove that a n =n 2 for all positive integers n Solution a1=1 a2=4 by induction a n-1 =(n-1) 2 a n =n 2 a n+1 =2a n -a n-1 +2 (from the sequence) putting the induced value into the sequence. a n+1 =2*n 2 -(n-1) 2 +2 =n 2 +2n +1=(n+1) 2 so a n =n 2 hence proved .