By dragging statements from the left column to the right column below, give a proof by induction of the following statement: For all n 1, 12 +22++n = n(n + 1)(2n+1) 6 The correct proof will use 8 of the statements below. Statements to choose from: Note that 12= (1)(1+1)(2-1+1) 6 So the base case P(1) is true. Now assume that P(k) is true for an arbitrary integer k 1. Let P(n) be the statement, 12+22+ n(n + 1)(2n+1). +17 = 6 Then we see that 12+22++(k+1) k(k + 1)(2k+1) = +(k+1) 6 k(k + 1)(2k + 1) 6(k + 1) = + 6 6 = = = 2k3+3k2+k 6k + 12k +6 6 2k3 +9k2+13k+6 6 (k+1)(k + 2)(2k +3 6 6 (k+1)((k+1)+1)(2(k+1)+1) 6 Note that 12 +22++(k+1) = (12+22++k)+(k+1). Then 12 +2 ++k = k(k + 1)(2k+1) 6 Therefore, by the Principle of Mathematical Induction, P(n) is true for all n 1. Thus P(k+1) is true. Your Proof: Put chosen statements in order in this column and press the Submit Answers button.