## Algebra and Trigonometry 10th Edition

Published by Cengage Learning

# Chapter 11 - 11.4 - Mathematical Induction - 11.4 Exercises - Page 806: 26

#### Answer

The inequality was proved for $n=1$ The inequality is correct if $n$ is changed by $n+1$

#### Work Step by Step

Let's prove the inequality for $n=7$: $(\frac{4}{3})^7\gt7$ $\frac{16384}{2187}\approx7.49\gt7$ It is correct! Now, suppose that the inequality is correct, that is: $(\frac{4}{3})^n\gt n$ Now, let's prove the inequality for $n+1$: $(\frac{4}{3})^{n+1}=(\frac{4}{3})(\frac{4}{3})^n=(1+\frac{1}{3})(\frac{4}{3})^n=(\frac{4}{3})^n+\frac{(\frac{4}{3})^n}{3}$ Since $...(\frac{4}{3})^9\gt(\frac{4}{3})^8\gt(\frac{4}{3})^7=\frac{16384}{2187}\approx7.49\gt3$ we have that: $\frac{(\frac{4}{3})^n}{3}\gt1$ if $n\geq7$: $(\frac{4}{3})^{n+1}=(\frac{4}{3})^n+\frac{(\frac{4}{3})^n}{3}\gt(\frac{4}{3})^n+1\gt n+1$ $(\frac{4}{3})^{n+1}\gt n+1$ That is exactly the given inequality if $n$ is changed by $n+1$.

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