Prove the following statement using mathematical induction. Do not derive it from Theorem 1 or Theorem 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.
P(1) = 1
1 + (5 · 1 − 4) = 1 · (5 · 1 − 3)
1 = 1 · (5 · 1 − 3)/2
P(1) = 1 · (5 · 1 − 3)/2
The selected statement is true because both sides of the equation equal . 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, and the right-hand side of P(k) is___.
[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) =. After substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomes+ (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. Hence P(k + 1) is true, which completes the inductive step.