## About Prove each of the following statements using mathematical induction. (a) Prove that for any positive integer n, 4 evenly divides 32n-1

Question

About Prove each of the following statements using mathematical induction. (a) Prove that for any positive integer n, 4 evenly divides 32n-1. (b) Prove that for any positive integer n, 6 evenly divides 7n – 1. (c) Prove that for any positive integer n, 4 evenly divides 11n – 7n. (d) Prove that for any positive integer n, 7 evenly divides 9n – 2n.

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5 months 2021-08-29T20:51:17+00:00 1 Answers 0 views 0

## Answers ( )

1. Answer:

(a)

(b)

(c)

(d)

Step-by-step explanation:

Solving (a): For integer , 4 divides

The proof is as follows:

For n = k, we have:

Multiply through by 4

Next, prove the statement is true for

Express -1 as – 9 + 8

Factorize:

Recall that:

So, we have:

Factorize

Since the above mathematical statement is true, then the given statement has been proved

Solving (b): For integer , 6 divides

The proof is as follows:

For n = k, we have:

Multiply through by 6

Next, prove the statement is true for

Express -1 as – 7 + 6

Factorize:

Recall that:

So, we have:

Factorize

Since the above mathematical statement is true, then the given statement has been proved

Solving (c): For integer , 4 divides

The proof is as follows:

For n = k, we have:

Multiply through by 4

Next, prove the statement is true for

Collect Like Terms

Recall that:

So, we have:

Factorize

Since the above mathematical statement is true, then the given statement has been proved

Solving (d): For integer , 7 divides

The proof is as follows:

For n = k, we have:

Multiply through by 7

Next, prove the statement is true for

Collect Like Terms

Recall that:

So, we have:

Factorize

Since the above mathematical statement is true, then the given statement has been proved