induction' j with j ih
j : ℕ
⊢ sum (fun i => i + i / 2 + 1) (j + j) = 3 * (j * j + j) + 1
case zero

⊢ sum (fun i => i + i / 2 + 1) (Nat.zero + Nat.zero) = 3 * (Nat.zero * Nat.zero + Nat.zero) + 1

case succ
j : ℕ
ih : sum (fun i => i + i / 2 + 1) (j + j) = 3 * (j * j + j) + 1
⊢ sum (fun i => i + i / 2 + 1) (Nat.succ j + Nat.succ j) = 3 * (Nat.succ j * Nat.succ j + Nat.succ j) + 1