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STD 11 - 12. limits — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 12. limits4 Marks
Question
Let $a_1, a_2, a_3 \ldots a_n$ be $n$ positive consecutive terms of an arithmetic progression. If $d > 0$ is its common difference, then $\lim _{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\ldots \ldots .+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}\right)$
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Answer

a
$\lim _{n \rightarrow \infty} \sqrt{\frac{ d }{ n }}\left(\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\ldots \ldots \ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}\right)$

On rationalising each term

$\lim _{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left(\frac{\sqrt{a_n}-\sqrt{a_1}}{d}\right)$

$\lim _{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left(\frac{(n-1) d}{\left(\sqrt{a_n}+\sqrt{a_1}\right) d}\right)=1$

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