For what natural numbers is the following inequality true? $( 1.15 ) ^ { n } n$
A) $n \geq 25$
B) $n \geq 2$
C) $n \geq 23$
D) $n \leq 23$
E) $37 \leq n \leq 54$

Question

For what natural numbers is the following inequality true? $( 1.15 ) ^ { n } > n$
A) $n \geq 25$
B) $n \geq 2$
C) $n \geq 23$
D) $n \leq 23$
E) $37 \leq n \leq 54$

Quizplus · Accepted Answer

We can use trial and error to find the answer. Evaluating the given inequality for various natural numbers, we get:
\begin{align*}
(1.15)^1 & > 1\
(1.15)^2 & > 2\
(1.15)^3 & > 3\
&\vdots\
(1.15)^{22} & > 22\
(1.15)^{23} & \approx 22.998 > 23\
(1.15)^{24} & \approx 26.398 > 24\
&\vdots
\end{align*}
Therefore, the inequality holds true for $n \geq 23$. Among the given answer choices, only option C lists this range of values.

For What Natural Numbers Is the Following Inequality True (1.15)n>n( 1.15 ) ^ { n } > n(1.15)n>n

For What Natural Numbers Is the Following Inequality True $( 1.15 ) ^ { n } > n$