- #1

Entertainment Unit

- 16

- 1

## Homework Statement

Show that if ##0 \leq a < b##, then

$$\frac {b^{n + 1} - a^{n + 1}} {b - a} < (n + 1)b^n$$

## Homework Equations

None that I'm aware of.

## The Attempt at a Solution

Proof (Induction)

1. Basis Case: Suppose ##n = 1##. It follows that:

$$\frac {b^{1 + 1} - a^{1 + 1}} {b - a} < (1 + 1)b^1$$

$$\frac {b^2 - a^2} {b - a} < 2b$$

$$b^2 - a^2< 2b(b - a)$$

$$b^2 - a^2< 2b^2- 2ab$$

$$-a^2< 2b^2$$

which is true for all ##0 \leq a < b##.

2. Inductive Step. Let ##k = n \geq 1## and suppose that:

$$\frac {b^{k + 1} - a^{k + 1}} {b - a} < (k + 1)b^k$$

This is where I can't make any progress. I can get the right-hand side to read ##(k + 2)b^{k + 1}## but doing so leaves the left-hand side of the inequality in a state that I can't get it to ##\frac {b^{k + 2} - a^{k + 2}} {b - a}##.