Concept explainers
Fill in the missing pieces in the following proof that
Proof: Let the property P(n) be the equation
Show that P(1) is true: To establish P(l), we must show that when 1 is substituted in place of n, the left-hand side equals the right-hand side. But when
Show that for every integer
[Suppose P(k) is true. That is:] Suppose
[This is the inductive hypothesis.] [We must show that
Now the left-hand side of
which is the right-hand side
[Since we have proved the basis step and the inductive step, we conclude that the given statement is true.]
Note: This proof was annotated to help make its logical flow more obvious. In standard mathematical writing, such annotation is omitted.
Prove each statement in 6—9 using mathematical induction. Do not derive them from Theorem 5.2.1 or Theorem 5.2.2.
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Discrete Mathematics With Applications
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,