Question

Sol. [Key Concept: Establish the inequality / identity.]

(a) $(1 \cdot 3 \cdot 5\ldots(2n - 1))^{1/n} < \frac{1 + 3 + 5 + \ldots(2n - 1)}{n} = n \Rightarrow 1 \cdot 3 \cdot 5\ldots(2n - 1) < n^{n}$

(b) $\ \frac{1 + 2 + 2^{2} + \ldots + 2^{n - 1}}{n} > \left( 2^{1 + 2 + \ldots(n - 1))^{1/n}} = 2^{\frac{n - 1}{2}} \right.\ $

$$\Rightarrow \ 2^{n} \cdot > 1 + n2^{n}$$

(c) $\frac{1}{n + 1} + \frac{1}{n + 2} + \ldots + \frac{1}{n + n} > \frac{1}{2n} + \frac{1}{2n} + \ldots + \frac{1}{2n} = \frac{1}{2}$

✔ (a) $n^{n} > 1 \cdot 3\cdots(2n - 1)$ by AM–GM

✔ (b) $2^{n} > 1 + n \cdot 2^{n - 1}$

✔ (c) $\sum_{k = n + 1}^{2n}\frac{1}{k} > \frac{12}{}$

✘ (d) reverse inequality is false