Prove that, for all natural numbers n, 1 + 4 + 9 + ...... +2 - n(n+1)(2n+1) 6 Select the steps necessary for a proof by induction. When n = 1, 1 +4 +9 + ...... +n- = 1, and O n(n+1)(2n+1) _1(2)(3) 6 = 1. 6 Assume the formula holds when n = k. 0 1+4+9+ ......+k2 _ k(k+1)(2k+1) 6 Since the result is true for n = 1 and the k + 1 case is true assuming the O k th case is true the formula holds for all cases. Since the k + 1 case is true assuming the k th case is true the formula 0 holds for all cases. Replace 1 + 4 + 9 + ...... +k2 with k(k+1)(2k+1) in 6 1+4+9+ ...... +k2+ (k+ 1)2 to show that O 1 + 4+9+ ...... +k2+ (k+ 1)2 - (*+ 1) (k + 2)(2k+ 3) 6 Prove the formula holds when n = k. 0 1 +4+9+ ...... +k2 _ k(k+1)(2k+1) 6 Maple TA