Let be positive consecutive terms of an arithmetic progression. If is its common difference, then:

1 Explanation: \begin{aligned} & \lim_{n\to\infty}\sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}}+\sqrt{a_{3}}+....+\frac{1}{\sqrt{a_{n+1}}+\sqrt{a_{n}}}\right) \\ & \text{一}\Rightarrow\operatorname*{lim}_{n\to\infty}\sqrt{\frac{d}{n}}\left(\frac{\sqrt{a_{1}}-\sqrt{a_{2}}}{a_{1}-a_{2}}+\frac{\sqrt{a_{2}}-\sqrt{a_{3}}}{a_{2}-a_{3}}+...+\frac{\sqrt{a_{n-1}}-\sqrt{a_{n}}}{\sqrt{a_{n}}+\sqrt{a_{n}}}\right) \\ & \Rightarrow\lim_{n\to\infty}\sqrt{\frac{d}{n}}\left(\frac{\sqrt{a_{n}}-\sqrt{a_{1}}}{d}\right) \\ & \Rightarrow\lim_{n\to\infty}\frac{1}{\sqrt{n}}\left(\frac{\sqrt{a+(n-1)d}-\sqrt{a_{1}}}{\sqrt{d}}\right) \\ & =1 \end{aligned}

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $a_{1}, a_{2}, a_{3}, ..., a_{n}$ be $n$ positive consecutive terms of an arithmetic progression. If $d > 0$ is its common difference, then: $n \to \infty lim n (\frac{1}{a _{1} + a _{2}} + \frac{1}{a _{2} + a _{3}} + ... + \frac{1}{a _{n - 1} + a _{n}})$

JEE 2023 — Mathematics PYQ

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