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Chapter 12: Q 24. (page 830)

In Problem, prove each statement.
If $0 < x < 1$ , then $0 < x^{n} < 1$

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 $0 < x^{1} < 1$ is true as this the condition given on $x$ , condition holds.

Next we assume that the statement is true for some $k$ , that is $0 < x^{k} < 1$

Now consider localid="1647761100625" $x^{k + 1} = x^{k} x$

Since

$0 < x^{k} < 1$ And

$0 < x < 1$

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

In Problems 61–70, express each sum using summation notation.

$1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + . . . . . . + {(- 1)}^{6} \frac{1}{3^{6}}$

In Problems 17–28, write down the first five terms of each sequence.

$\{t_{n}\} = \{\frac{{(- 1)}^{n}}{(n + 1 (n + 2)}\}$

How much do you need to invest now at $5 %$ per annum compounded monthly so that in 1 year you will have $$ 10000$ ?

Education IRA: On January 1, 1999, John’s parents decided to place \(45 at the end of each month into an Education IRA.

(a) Find a recursive formula that represents the balance at the end of each month if the rate of return is assumed to be 6% per annum compounded monthly.

(b) How long will it be before the value of the account exceeds \)4000?

(c) What will be the value of the account in 16 years when John goes to college?

In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.

$1 + \frac{1}{3} + \frac{1}{9} + . . . . .$

See all solutions

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