Question

If $a_1,a_2,.......a_n$ are in arithmetic progression with common difference $d>n$, then find limit $\lim_{n \to \infty}\sqrt{\frac{d}{n}}(\frac{1}{\sqrt{a_1}+\sqrt{a_2}})+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+.......+\frac{1}{\sqrt{a_n}+\sqrt{a_{n-1}}})$ is____.

Answer 1

$\lim_{n \to \infty}\sqrt{\frac{d}{n}}(\frac{1}{\sqrt{a_2}-\sqrt{a_1}}+\frac{1}{\sqrt{a_3}-\sqrt{a_2}}+........+\frac{1}{\sqrt{a_n}-\sqrt{a_{n-1}}})$
$=\frac{1}{d} \sqrt{\frac{d}{n}}(\sqrt{a_n}-\sqrt{a_1})$
$= \frac{1}{d}\sqrt{\frac{d}{n}}(a_1+nd)^\frac{1}{2}$
$= \frac{1}{d}\sqrt{\frac{d}{n}}\sqrt{n}(d+\frac{a_1}{n})^{\frac{1}{2}}$
$= \frac{1}{d}\sqrt{d}\times\sqrt{d}(1+\frac{a_1}{nd})^{\frac{1}{2}}$
$=1$