Use mathematical induction to prove that the statement is true for every positive integer n.1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + n(n + 1) = 
What will be an ideal response?
Answers will vary. One possible proof follows.
a). Let n = 1. Then, 1?2 = 2 = = 2. So, the statement is true for n = 1.
b). Assume the statement is true for n = k:
Sk =
Also, if the statement is true for n = k + 1, then
Sk+1 = Sk + (k + 1)(k + 2) = .
Subtracting, we get:
Sk+1 - Sk = (k + 1)(k + 2) = -
.
Expand both sides and collect like terms:
k2 + 3k + 2 = -
=
= k2 + 3k + 2
Since the equality holds, then the statement is true for n = k + 1 as long as it is true for n = k. Furthermore, the statement is true for n = 1. Therefore, the statement is true for all natural numbers n.
You might also like to view...
Find the zeros of the function. Give exact answers.f(x) = x2 - 5x - 6
A. -1, 6 B. 1, 6 C. 1, -6 D. -1, -6
Solve the quadratic equation by completing the square.3x2 - 6x = 45
A. -15, 1 B. -3, 15 C. 3, -5 D. -3, 5
Perform the indicated operations and simplify the result. Leave the answer in factored form.
A. -
B. -
C. -1
D.
Solve the problem.A ladder is slipping down a vertical wall. If the ladder is 20 ft long and the top of it is slipping at the constant rate of 5 ft/s, how fast is the bottom of the ladder moving along the ground when the bottom is 16 ft from the wall?
A. 0.8 ft/s B. 3.8 ft/s C. 0.31 ft/s D. 6.3 ft/s