Find the smallest value of N that will make the inequality hold for all n > N.
- $| \sqrt [ n ] { 2 n } - 1 | < 10 ^ { - 3 }$

Question

Find the smallest value of N that will make the inequality hold for all n > N.
-$$| \sqrt [ n ] { 2 n } - 1 | < 10 ^ { - 3 }$$
A) 9883
B) 9891
C) 9899
D) 9907

Quizplus · Accepted Answer

The inequality $$| \sqrt [ n ] { 2 n } - 1 | < 10 ^ { - 3 }$$ can be simplified by considering the approximation for large $n$, where $\sqrt[n]{2n}$ approaches 1. To find the smallest $N$, we need to solve the inequality for $n$. First, we can rewrite the inequality without the absolute value by considering the positive and negative cases separately, but since $\sqrt[n]{2n}$ is always greater than 1 for $n > 1$, we only need to consider the case where $\sqrt[n]{2n} - 1 < 10^{-3}$.Rearranging gives us $\sqrt[n]{2n} < 1 + 10^{-3}$. Raising both sides to the power of $n$ gives $2n < (1 + 10^{-3})^n$. To find the smallest $n$ that satisfies this, we can test the given options directly (since solving explicitly for $n$ in terms of elementary functions is not feasible):- For $n = 9883$, the inequality does not hold.- For $n = 9891$, the inequality does not hold.- For $n = 9899$, the inequality holds.- For $n = 9907$, the inequality also holds, but we are looking for the smallest $n$.Therefore, the smallest value of $N$ that makes the inequality hold for all $n > N$ is 9899.

Find the Smallest Value of N That Will Make the Inequality