The value of the greatest integer k satisfyin | vedclass

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(B) Let $m = n+4$. Since $n \in \mathbb{N}$,$n \geq 1$,so $m \geq 5$.The given inequality becomes $2^m + 12 \geq km$,which implies $k \leq \frac{2^m +

Vedclass · Accepted Answer

$8$

Vedclass · Answer

(B) Let $m = n+4$. Since $n \in \mathbb{N}$,$n \geq 1$,so $m \geq 5$.The given inequality becomes $2^m + 12 \geq km$,which implies $k \leq \frac{2^m + 12}{m}$ for all $m \geq 5$.To find the greatest integer $k$,we need to find the minimum value of the function $f(m) = \frac{2^m + 12}{m}$ for $m \in \{5, 6, 7, \dots\}$.For $m = 5$: $f(5) = \frac{2^5 + 12}{5} = \frac{32 + 12}{5} = \frac{44}{5} = 8.8$.For $m = 6$: $f(6) = \frac{2^6 + 12}{6} = \frac{64 + 12}{6} = \frac{76}{6} \approx 12.66$.For $m = 7$: $f(7) = \frac{2^7 + 12}{7} = \frac{128 + 12}{7} = \frac{140}{7} = 20$.As $m$ increases,$f(m)$ increases for $m \geq 5$.Thus,the minimum value is $8.8$ at $m = 5$.Since $k \leq f(m)$ for all $m$,$k$ must be less than or equal to the minimum value of $f(m)$.Therefore,$k \leq 8.8$.The greatest integer $k$ is $8$.

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

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