Prove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2.
For every integer n ≥ 1,
1 + 6 + 11 + 16 + ... + (5n - 4) = n(5n - 3)/2.
Proof (by mathematical induction):
Let P(n) be the equation
1 + 6 + 11 + 16 + ... + (5n - 4) = n(5n - 3)/2.
We will show that P(n) is true for every integer n ≥ 1.
Show that P(1) is true:
Select P(1) from the choices below.
1 + (5 · 1 - 4) = 1 · (5 · 1 - 3)/2
1 = 1 · (5 · 1 - 3)/2
P(1) = 5 · 1 - 4
P(1) = 1 · (5 · 1 - 3)/2
The selected statement is true because both sides of the equation equal each other.
Show that for each integer k ≥ 1, if P(k) is true, then P(k + 1) is true:
Let k be any integer with k ≥ 1, and suppose that P(k) is true. The left-hand side of P(k) is 5k - 4, and the right-hand side of P(k) is k(5k - 3)/2. [The inductive hypothesis states that the two sides of P(k) are equal.]
We must show that P(k + 1) is true. P(k + 1) is the equation 1 + 6 + 11 + 16 + ... + (5(k + 1) - 4) = k(5k - 3)/2 + (5(k + 1) - 4). After substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomes k(5k - 3)/2 + (5(k + 1) - 4). When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal (k + 1)(5(k + 1) - 3)/2. Hence P(k + 1) is true, which completes the inductive step.
[Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]