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ISI 2021 | PCB Mathematics | Question: 10

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Let $a>0$ and $x_{1}>0$. Define $x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right)$ for all $n \in \mathbb{N}$. Show that

  1. $x_{n}>\sqrt{a}$ for all $n \geq 2;$
  2. the sequence $\left\{x_{n}: n \geq 1\right\}$ converges to $\sqrt{a}.$
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ISI 2021 | PCB Mathematics | Question: 2
Consider three real numbers $a \geq b \geq c>0$. If $\left(a^{x}-b^{x}-c^{x}\right)(x-2)>0$ for any rational number $x \neq 2$, show that $a, b$ and $c$ can be the lengths of the three sides of a triangle $A B C;$ $A B C$ is a right-angled triangle.
Consider three real numbers $a \geq b \geq c>0$. If $\left(a^{x}-b^{x}-c^{x}\right)(x-2)>0$ for any rational number $x \neq 2$, show that$a, b$ and $c$ can be the lengths...
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