chapter 5

Put these steps in order to complete the inductive step of the proof by mathematical induction that 2n< n! for every integer n with n≥4.

1. let 2. assume 3. we must 4. to begin 5. using 6. therefore

Match these components of a proof that P(n) is true for all positive integers n with their definitions: Instructions

Basis step matches Choice Verify that P(1) is true. Inductive hypothesis matches Choice The assumption that P(k) is true for some arbitrary positive integer k. Inductive step matches Choice Show that for all positive integers k, if P(k) is true then P(k + 1) is true.

Which of the these are steps for a proof by mathematical induction that P(n) is true for all positive integers n

Verify that P(1) is true. Demonstrate that the conditional statement P(k) implies P(k+1) is true for all positive integers k.

Which of these steps are part of a template for proofs by mathematical induction?

show that P(b) is true. Show that if P(k) is true for an arbitrary fixed integer k ≥ b, then P(k + 1) is true. Express the statement to be proved in the form "for all n ≥ b, P(n)" for a fixed integer b. Write "By mathematical induction, P(n) is true for all integers n with n ≥ b."