If n N ( n+1 2 )^n ! is true when: - Vidyadip

MCQ

If $n \in N \Big(\frac{\text{n}+1}{2}\Big)^\text{n}\geq\text{n!}$ is true when:

✓
$n ≥ 1$
B
$n ≥ 2$
C
$n > 1$
D
$n > 2$

✓

Answer

Correct option: A.

$n ≥ 1$

Concept:
$\text{n!}=\text{n}\times(\text{n}-1)\times\text{n}-2.....\times3\times2\times1$
Calculation:
Given:
$\text{p}\text{(n)}=\Big(\frac{\text{n+1}}{2}\Big)^\text{n}\geq\text{n!}$
Put $n = 1$
$\text{p}(2)=\Big(\frac{2+1}{2}\Big)^2\geq2!$
$=\Big(\frac{3}{2}\Big)^2\geq2\times1$
$2.25 ≥ 2$
Put $n = 3$
$\text{p}(3)=\Big(\frac{3+1}{2}\Big)^3\geq3!$
$8\geq3\times2\times1,8\geq6$
Hence, The given expression $P(n)$ is true for $n ≥ 1$

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