Mathematical Induction and Divisibility

This video tutorial focuses on solving induction divisibility problems. It provides step-by-step proofs for three mathematical expressions: proving that 6^n + 4 is divisible by 5, 7^n - 2^n is divisible by 5, and n(n+2) is divisible by 4 for even integers. The tutorial explains the process of mathematical induction, including the base case, induction hypothesis, and induction step, to demonstrate the divisibility of these expressions.

What is the first step in proving a statement using mathematical induction?

• Assume the statement is false for n = 1

• Prove the statement is true for n = 1

• Assume the statement is true for n = k

• Prove the statement is true for n = k + 1

In the expression 6^n + 4, what is the base case for proving divisibility by 5?

• n = 1

• n = 3

• n = 2

• n = 0

What is the purpose of the induction step in mathematical induction?

• To assume the statement is true for n = k

• To find a counterexample

• To prove the statement for n = 1

• To prove the statement for n = k + 1

What expression do we need to prove is divisible by 5 in the second problem?

• 6^n + 4

• 7^n - 2^n

• 5^n - 3^n

• n(n + 2)

In the expression 7^n - 2^n, what is the base case for proving divisibility by 5?

• n = 0

• n = 3

• n = 1

• n = 2

What is the greatest common factor used in the proof of 7^n - 2^n?

• 5

• 2

• 7

• 3

What is the expression to prove divisibility by 4 in the third problem?

• 5^n - 3^n

• n(n + 2)

• 7^n - 2^n

• 6^n + 4

For which type of integers is the expression n(n + 2) divisible by 4?

• All odd integers

• All positive even integers

• All negative integers

• All positive integers

What is the first positive even integer used in the base case for n(n + 2)?

• 3

• 4

• 2

• 1

What is the induction step for proving n(n + 2) is divisible by 4?

• Assume false for n = k and prove for n = k + 2

• Assume false for n = k and prove for n = k + 1

• Assume true for n = k and prove for n = k + 1

• Assume true for n = k and prove for n = k + 2