IIT-JEE 1981Question 1
State TRUE or FALSE: For every integer $n>1$, the inequality $(n!)^{1/n}<\dfrac{n+1}{2}$ holds.
Sequence and Series · Means and Inequalities (AM, GM, HM)
Solution
Formulas used. (1) AM-GM: $\frac{x_1+\cdots+x_n}{n}>\left(x_1\cdots x_n\right)^{1/n}$ when the numbers are not all equal(2) $1+2+\cdots+n=\frac{n(n+1)}{2}$Step 1 - Apply AM-GM to $1,2,\dots,n$ (not all equal, since $n>1$): $$\left(1\cdot2\cdots n\right)^{1/n}<\frac{1+2+\cdots+n}{n}=\frac{n+1}{2}.$$ Step 2. So $(n!)^{1/n}<\dfrac{n+1}{2}$ for every $n>1$: TRUE. Answer: (A)