Using mathematical induction, the numbers a n 's are defined by a 0 = 1 , a n + 1 = 3 n 2 + n + a n , n ≥ 0 , then a n =

Step-by-step solution

Given, a 0 = 1 , a n + 1 = 3 n 2 + n + a n   n ≥ 0 Putting, n = 0 , a 0 + 1 = 3 × 0 + 0 + a 0 a 1 = a 0 = 1 n = 1 , a 1 + 1 = 3 × 1 2 + 1 + a 1 a 2 = 3 + 1 + 1 = 5 n = 2 . a 2 + 1 = 3 2 2 + 2 + 5 a 3 = 19 a n :   a 3 1   n 3 + n 2 + 1   ⇒ 3 3 + 3 2 + 1 ⇒ 37   , (wrong option) 2   n 3 - n 2 + 1 ⇒ 3 3 - 3 2 + 1 ⇒ 19   , (right option) ( 3 )   n 3 - n 2 ⇒ 3 3 - 3 2 = 18   (wrong option) 4   n 3 + n 2 ⇒ 3 3 + 3 2 &#865