n^n ( n + 1 2 )^ 2n is - Vidyadip

MCQ

${n^n}{\left( {\frac{{n + 1}}{2}} \right)^{2n}}$ is

A
Less than ${\left( {\frac{{n + 1}}{2}} \right)^3}$
B
Greater than ${\left( {\frac{{n + 1}}{2}} \right)^3}$
C
Greater than ${(n!)^3}$
✓
$(b)$ and $(c)$ both

✓

Answer

Correct option: D.

$(b)$ and $(c)$ both

d
(b) $y = {n^n}{\left( {\frac{{n + 1}}{2}} \right)^{2n}}$

Put $n = 2$, $y = {2^2}{\left( {\frac{3}{2}} \right)^4} = 4\,.\,\frac{{81}}{{8 \times 2}} = \frac{{81}}{4}\tilde - 20$

Option $(a) = {\left( {\frac{{n + 1}}{2}} \right)^3} = \frac{{27}}{8} < y$

Option $(b) = {\left( {\frac{{n + 1}}{2}} \right)^3} = \frac{{27}}{8} < y$

Option $(c) = {(2!)^3} = 8 < y$

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