Question

If $x>0$ then greatest value of the expression $\frac{x^{50}}{1 + x + x^{2} + \ldots \ldots + x^{100}}$ is

• A. $\frac{1}{102}$
• B. $\frac{1}{101}$
• C. $\frac{1}{100}$
• D. none of these

Answer 1

Answer: (B)

$A.M.\geq G.M.$
$\Rightarrow \frac{1 + x + x^{2} + \ldots . . + x^{100}}{101}\geq \left(1 . x \cdot x^{2} \ldots \ldots . . x^{100}\right)^{\frac{1}{101}}$
$\Rightarrow \frac{1 + x + x^{2} + \ldots \ldots . . + x^{100}}{101}\geq x^{\frac{100 \times 101}{2} \times \frac{1}{101}}$
$\Rightarrow \frac{1 + x + x^{2} + \ldots \ldots . . . x^{100}}{101}\geq x^{50}$
$\Rightarrow \frac{1}{101}\geq \frac{x^{50}}{1 + x + x^{2} + \ldots \ldots + x^{100}}$
$\therefore $ Greatest value $=\frac{1}{101}$