Find the smallest value of N that will make the inequality hold for all n > N.
- $| \sqrt [ n ] { 0.8 } - 1 | < 10 ^ { - 2 }$

Question

Find the smallest value of N that will make the inequality hold for all n > N.
-$$| \sqrt [ n ] { 0.8 } - 1 | < 10 ^ { - 2 }$$
A) 25
B) 23
C) 27
D) 20

Quizplus · Accepted Answer

First, we can simplify the inequality as follows:
$-\frac{1}{100}<\sqrt[n]{0.8}-1<\frac{1}{100}$

Adding 1, we get:

$\frac{99}{100}<\sqrt[n]{0.8}<\frac{101}{100}$

Taking the $n$-th power of each side, we get:

$\frac{99^n}{100^n}<0.8<\frac{101^n}{100^n}$

Simplifying this inequality, we get:

$5^n<125<\frac{25^n}{2^n}$

Since $\frac{25^n}{2^n}=5^nn$, this inequality is equivalent to:

$5^n<125<5^nn$

Dividing through by $5^n$, we get:

$1<\frac{125}{5^n}<n$

$1<5^{3-n}<n$

Since $5^{3-n}$ is increasing with $n$, we want to find the smallest $n$ for which $5^{3-n}\geq n$. We can easily try different values of $n$ to find that $n=23$ is the smallest value that works. Therefore, the answer is $\boxed{	extbf{(B) }23}$.

Find the Smallest Value of N That Will Make the Inequality