Register

ISI2017-MMA-7

• edited by
318 views
0 votes
0 votes

Let $n \geq 3$ be an integer. Then the statement $(n!)^{1/n} \leq \dfrac{n+1}{2}$ is

  1. true for every $n \geq 3$
  2. true if and only if $n \geq 5$
  3. not true for $n \geq 10$
  4. true for even integers $n \geq 6$, not true for odd $n \geq 5$
• edited by
User Avatar

1 Answer

Best answer
2 votes
2 votes
By AM-GM inequality, We have $$\sqrt[n]{1\cdot2\cdot3\cdot...\cdot n}\le\frac{1+2+3+...+n}{n}=\frac{n(n+1)}{2n}=\frac{n+1}{2}$$

$\Rightarrow ( n!)^{\frac{1}{n}}\leq \frac{n+1}{2}$

Hence, Option $(A)$ is correct.
• selected by
User Avatar
Voting & Accepting an answer helps improve the community
← PreviousNext →
← Previous in categoryNext in category →

Related questions

0 votes
0 votes
0 answers
1
go_editor asked Sep 15, 2018
478 views
ISI2017-MMA-3
If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+3x^2-8x+1=0$, then an equation whose roots are $\alpha+1, \beta+1$ and $\gamma+1$ is given by $y^3-11y+11=0$ $y^3-11y-11=0$ $y^3+13y+13=0$ $y^3+6y^2+y-3=0$
If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+3x^2-8x+1=0$, then an equation whose roots are $\alpha+1, \beta+1$ and $\gamma+1$ is given by $y^3-11y+...
User Avatar
0 votes
0 votes
1 answer
2
go_editor asked Sep 15, 2018
328 views
ISI2017-MMA-1
The area lying in the first quadrant and bounded by the circle $x^2+y^2=4$ and lines $x=0 \text{ and } x=1$ is given by $\frac{\pi}{3}+\frac{\sqrt{3}}{2}$ $\frac{\pi}{6}+\frac{\sqrt{3}}{4}$ $\frac{\pi}{3}-\frac{\sqrt{3}}{2}$ $\frac{\pi}{6}+\frac{\sqrt{3}}{2}$
The area lying in the first quadrant and bounded by the circle $x^2+y^2=4$ and lines $x=0 \text{ and } x=1$ is given by$\frac{\pi}{3}+\frac{\sqrt{3}}{2}$$\frac{\pi}{6}+\f...
User Avatar
0 votes
0 votes
1 answer
3
go_editor asked Sep 15, 2018
555 views
ISI2017-MMA-9
A function $y(x)$ that satisfies $\dfrac{dy}{dx}+4xy=x$ with the boundary condition $y(0)=0$ is $y(x)=(1-e^x)$ $y(x)=\frac{1}{4}(1-e^{-2x^2})$ $y(x)=\frac{1}{4}(1-e^{2x^2})$ $y(x)=\frac{1}{4}(1-\cos x)$
A function $y(x)$ that satisfies $\dfrac{dy}{dx}+4xy=x$ with the boundary condition $y(0)=0$ is$y(x)=(1-e^x)$$y(x)=\frac{1}{4}(1-e^{-2x^2})$$y(x)=\frac{1}{4}(1-e^{2x^2})$...
User Avatar
0 votes
0 votes
1 answer
4
admin asked May 21, 2023
375 views
GATE Mechanical 2023 | GA Question: 7
Consider the following inequalities $ \begin{aligned} & p^2-4 q<4 \\ & 3 p+2 q<6 \end{aligned} $ where $p$ and $q$ are positive integers. The value of $(p+q)$ is __________. $2$ $1$ $3$ $4$
Consider the following inequalities$$\begin{aligned}& p^2-4 q<4 \\& 3 p+2 q<6\end{aligned}$$where $p$ and $q$ are positive integers.The value of $(p+q)$ is __________.$2$...
User Avatar
Link Copied!