A sequence $x_{1}, x_{2}, x_{3}, . . . . . .$ is defined by letting $x_{1} = 2$ and $x_{k} = \frac{x_{k - 1}}{n}$ for all natural numbers k, $k \geq 2$ . Show that $x_{n} = \frac{2}{n!}$ for all $n ϵ N$ .

Q. The range of

a

for which

x^{2} - a x + 1 - 2 a^{2}

is always positive for all real values of

x

, is

Q. Determine all real vales of x for which the expression

{\frac{2}{x^{2} - x + 1} - \frac{1}{x + 1} - \frac{2 x + 1}{x^{3} + 1}}^{1 / 2}

takes real values.

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x3≥x2 for all positive numbers x.Which of the following values for x shows that the statement above is false?

The correct option is A 12x3>x2 for all posive numbers x. which givesx2×(x−1)≥0. Since x2≥0, (x−1) must be ≥0.Therefore x=1/2 shows that the given statement is wrong.

$x^{3} \geq x^{2}$ for all positive numbers x.
Which of the following values for x shows that the statement above is false?

The correct option is A $\frac{1}{2}$
$x^{3} > x^{2}$ for all posive numbers $x$ . which gives
$x^{2} \times (x - 1) \geq 0$ . Since $x^{2} \geq 0$ , $(x - 1)$ must be $\geq 0$ .
Therefore $x = 1 / 2$ shows that the given statement is wrong.