# To state: a general rule about the relationship between a n and b n when a &lt; b and n is any integer.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.7, Problem 122E
To determine

## To state: a general rule about the relationship between anand bn when a<b and n is any integer.

Expert Solution

The solution an<bn

### Explanation of Solution

Given:

If a<b, is a2<b2 any a<b, is a3<b3 .

Calculation:

According to the question, to check all positive and negative values,

Case 1: both a andb negative. Then:

a<ba2>b2{ifa<0,b<0}

Case 2: both anegative andbpositive . Then:

Case 3: both a andb positive. Then:

ifa>0,b>0a2<b2

Case 4: both a andb negative. Then:

a<ba3<b3{ifa<0,b<0}

Case 5: anegative andbpositive . Then:

a<0,b<0a3<b3

Case 6: both positive. Then:

a<0,b>0a3<b3

Case 7: From the above observation, an<bn for given a<b .

Case8: If n is even,

If a<0,b<0

Then an>bn

ifa>0,b>0

an<bn

Case9: If n is odd,

If a<0,b<0

Then an<bn

Ifa<0,b>0 an<bn

ifa>0,b>0an<bn

Hence, the relationship between anand bn is above solved.

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