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Chapter 12: Q 24. (page 830)

In Problem, prove each statement.

If0<x<1, then0<xn<1

Short Answer

Expert verified

The statement is shown.

Step by step solution

01

Step 1. Given information

0<x<1

02

Step 2. Prove that 0<xn<1.

First, we consider the statement for n=1

Because 0<x1<1is true as this the condition given on x, condition holds.

Next we assume that the statement is true for some k, that is 0<xk<1

Now consider localid="1647761100625" xk+1=xkx

Since

0<xk<1And

0<x<1

Multiplying these two, we have

width="109">0.0<xk.x<1.1

width="86">0<xk+1<1

So. condition also holds true.

As a result we say that the given statement is true for all natural numbers n

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Most popular questions from this chapter

A pond currently has 2000 trout in it. A fish hatchery decides to add an additional 20 trout each month. In addition, it is known that the trout population is growing 3% per month. The size of the population after n months is given by the recursively defined sequence.

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