# To prove: If n is positive integer, then 7 n − 1 is divisible by 6.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 1, Problem 17P
To determine

## To prove: If n is positive integer, then 7n−1 is divisible by 6.

Expert Solution

### Explanation of Solution

Proof:

Use mathematical induction on n.

Let Sn be the statement 7n1 is divisible by 6

Base case: n=1

To prove that the statement is true for n=1.

711=6

Clearly, it is divisible by 6.

Therefore, the statement is true for n=1.

Induction hypothesis: n=k

Assume that the claim is true for n=k.

That is, 7k1=6m, for some positive integer m

Thus 7k=6m+1.

Inductive step: n=k+1

To prove that the statement is true for n=k+1.

7k+11=7k71=(6m+1)71 [From induction hypothesis]=42m+6=6(7m+1)

Thus, the statement is true for n=k+1.

Therefore, by mathematical induction, the statement 7n1 is divisible by 6.

Hence the required result is obtained.

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