Question

The value of the greatest integer k satisfying the inequation $2^{n+4} + 12 \geq k(n+4)$ for all $n \in N$ is

• A. 7
• B. 8
• C. 9
• D. 10

Hint:

When an inequality of the form $k \leq f(n)$ must hold for all $n$ in a set, $k$ must be less than or equal to the minimum value of $f(n)$ over that set. To find the minimum, test the first few values or use calculus to analyze the function's behavior.

Answer 1

The Correct Option is B

The given inequation is $2^{n+4} + 12 \geq k(n+4)$ for all $n \in N$.
We can rearrange the inequality to solve for $k$:
$k \leq \frac{2^{n+4} + 12}{n+4}$.
This inequality must hold for all natural numbers $n$ (i.e., $n = 1, 2, 3, \dots$).
Therefore, $k$ must be less than or equal to the minimum value of the expression $\frac{2^{n+4} + 12}{n+4}$.
Let $f(n) = \frac{2^{n+4} + 12}{n+4}$. We need to find the minimum value of $f(n)$ for $n \in N$.
Let's test the first few values of $n$:
For $n = 1$: $f(1) = \frac{2^{1+4} + 12}{1+4} = \frac{2^5 + 12}{5} = \frac{32 + 12}{5} = \frac{44}{5} = 8.8$.
For $n = 2$: $f(2) = \frac{2^{2+4} + 12}{2+4} = \frac{2^6 + 12}{6} = \frac{64 + 12}{6} = \frac{76}{6} \approx 12.67$.
For $n = 3$: $f(3) = \frac{2^{3+4} + 12}{3+4} = \frac{2^7 + 12}{7} = \frac{128 + 12}{7} = \frac{140}{7} = 20$.
The values of $f(n)$ are increasing as $n$ increases. This indicates the minimum value of $f(n)$ occurs at the smallest value of $n$, which is $n = 1$.
The minimum value of the expression is $8.8$.
The condition becomes $k \leq \min(f(n))$, which is $k \leq 8.8$.
The question asks for the greatest integer value of $k$ that satisfies this condition.
The greatest integer less than or equal to $8.8$ is $8$.