Question

Assertion :For $n\in N$, ${(n!)}^3$ $<n^n{\left(\frac{n+1}{2}\right)}^{2n}$ Reason: $n>6,{\left(\frac{n}{3}\right)}^n<n!{\left(\frac{n}{2}\right)}^n$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Using $A.M>G.M$
$\frac{1^3+2^3+...+n^3}{n}>{(1^3\cdot 2^3\cdot 3^3...n^3)}^{1/n}$
$\Rightarrow \frac{\sum n^3}{n}>{(n!)}^{3/n}$
$\Rightarrow \frac{n^2{(n+1)}^2}{4n}>{(n!)}^{3/n}$
$\Rightarrow {\left[{\left(\frac{n(n+1)}{2}\right)}^2\right]}^n>{(n!)}^3\Rightarrow n^n{\left(\frac{n+1}{2}\right)}^{2n}>{(n!)}^3$
$\Rightarrow {(n!)}^3<n^n{\left(\frac{n+1}{2}\right)}^{2n}$
Hence Assertion (A) & Reason (R) both are true but Reason (R) is not proper.
Ans: B