Prove that for all natural numbers n, 1 + 4 + 16 + ...... +47-1 = (47 - 1) . Select the steps necessary for a proof by induction. When n = 1, 1 + 4 + 16 + ...... +47-1 = 1, -(4- 1) = 1. When n = 2, 1 + 4+ 16 + ...... +47-1 = 1+4 = 5; =(42 - 1) = =(4-1)(4 +1) = 1+4 = 5. When n = 3, 1 + 4+ 16 + ...... +47-1 = 1+4+16 = 21; (43 -1) = (4 - 1)(16 + 4 + 1) = 1+4+16 = 21, and so on. Assume the formula holds when n = k. 0 1+4 + 16+ ......+4k-1 = 4(4k - 1) Show that 1 + 4 + 16 + ...... +4k = 1(4k+1 - 1). When n = 1, 1 + 4 + 16 + ...... +47-1 = 1 and 2 (41 - 1) = 1. Assume 1 + 4 + 16 + ...... +47 = 1(4n+1 - 1). Since the k + 1 case is true assuming the k th case is true the formula O holds for all cases. 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. Maple T.A