What are the limits of 2 n n -1 n - n n- where n is a positve integer-A- 2 -2nn-1n-nn- 3B- 2n-1-2nn-1n-nn- 3-2nC- 2n-2nn-1n-nn- 3nD- N one of these

The correct option is B 2^n+1≤2^n (n + 1)^n/n^n≤ 3. 2^n2^n (n + 1)^n/n^n = 2^n (n+1/n)^n = 2^n (1 + 1/n)^n Now, we have 2 ≤ (1 + 1/n)^n ≤ 3 ∴ 2.2^n ≤ 2^n (1 + ...

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Quantitative Aptitude

Solving Inequalities

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Question

What are the limits of $\frac{2^{n} (n + 1)^{n}}{n^{n}}$ ; where n is a positve integer ?

2 \leq \frac{2^{n} (n + 1)^{n}}{n^{n}} \leq 3

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2^{n + 1} \leq \frac{2^{n} (n + 1)^{n}}{n^{n}} \leq 3. 2^{n}

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2^{n} \leq \frac{2^{n} (n + 1)^{n}}{n^{n}} \leq 3^{n}

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Solution

The correct option is B $2^{n + 1} \leq \frac{2^{n} (n + 1)^{n}}{n^{n}} \leq 3. 2^{n}$
$\frac{2^{n} (n + 1)^{n}}{n^{n}} = 2^{n} (\frac{n + 1}{n})^{n} = 2^{n} (1 + \frac{1}{n})^{n}$
Now, we have $2 \leq (1 + \frac{1}{n})^{n} \leq 3$
$∴ {2.2}^{n} \leq 2^{n} (1 + \frac{1}{n})^{n} \leq {3.2}^{n} i . e ., 2^{n + 1} \leq 2^{n} (1 + \frac{1}{n})^{n} \leq {3.2}^{n} .$

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What are the limits of 2n(n+1)nnn ; where n is a positve integer ?

The correct option is B 2n+1≤2n(n+1)nnn≤3.2n2n(n+1)nnn=2n(n+1n)n=2n(1+1n)n Now, we have 2≤(1+1n)n≤3 ∴2.2n≤2n(1+1n)n≤3.2ni.e.,2n+1≤2n(1+1n)n≤3.2n.

What are the limits of $\frac{2^{n} (n + 1)^{n}}{n^{n}}$ ; where n is a positve integer ?