Find the smallest value of N that will make the inequality hold for all n > N.
- $| \sqrt [ n ] { 0.1 } - 1 | < 10 ^ { - 3 }$

Question

Find the smallest value of N that will make the inequality hold for all n > N.
-$$| \sqrt [ n ] { 0.1 } - 1 | < 10 ^ { - 3 }$$
A) 2302
B) 2305
C) 2300
D) 2298

Quizplus · Accepted Answer

The inequality $| \sqrt[n]{0.1} - 1 | < 10^{-3}$ can be rewritten as $-10^{-3} < \sqrt[n]{0.1} - 1 < 10^{-3}$. This implies $0.999 < \sqrt[n]{0.1} < 1.001$. Taking logarithms, we get $\log_{10}(0.1)/n > \log_{10}(0.999)$ and $\log_{10}(0.1)/n < \log_{10}(1.001)$. Solving these inequalities for n gives the smallest integer n as 2302.

Find the Smallest Value of N That Will Make the Inequality