Which of the following hold good? If n is a +ve integer, then

n^{n} > 1.3.5 \dots \dots (2 n - 1)

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2.4.6 \dots \dots 2 n < (n + 1)^{n}

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

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[1^{r} + 2^{r} + 3^{r} + \dots \dots + n^{r}]^{n} > n^{n} . (n!)^{r}

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Solution

The correct options are
A $n^{n} > 1.3.5 \dots \dots (2 n - 1)$
B $2.4.6 \dots \dots 2 n < (n + 1)^{n}$
C $(n!)^{3} < n^{n} {(\frac{n + 1}{2})}^{2 n}$
D $[1^{r} + 2^{r} + 3^{r} + \dots \dots + n^{r}]^{n} > n^{n} . (n!)^{r}$
(a), (b), (c), (d)
(a) Apply A.M. of 1,3,5 .... (2n-1), i.e. of n numbers is > G.M. and remember that 1+3+5+... upto n terms $= \frac{n}{2} [2 a + (n - 1) d] = n^{2}$ as a=1 and d=2

(b) Proceed as in part (a)

(c) Apply A.M. of $1^{3}, 2^{3}, \dots \dots n^{3}$ is greater than their G.M. and remember that
$1^{3} + 2^{3} + \dots \dots n^{3} = {\frac{n (n + 1)}{2}}^{2}$

(d) $\frac{1^{r} + 2^{r} + 3^{r} + \dots \dots + n^{r}}{n} > [1^{r} . 2^{r} . 3^{r} . \dots \dots . n^{r}]^{\frac{1}{n}}$
or $(1^{r} + 2^{r} + 3^{r} + \dots \dots + n^{r})^{n} > n^{n} (n!)^{r}$

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