Verified Solution • 2025–26 Session moderate

QJEEM23Mabde9: JEE Main 2023 Mathematics - Class 12 JEEM23Mabde9 Solution: Let be positive consecutive terms of an arithmetic progression. …

Let a1,a2,a3,…,ana_{1}, a_{2}, a_{3}, \ldots, a_{\mathrm{n}} be n\mathrm{n} positive consecutive terms of an arithmetic progression. If \mathrm{d} > 0 is its common difference, then

limn→∞dn(1a1+a2+1a2+a3+………+1an−1+an) is

Multiple Choice Options

  • A 1d\frac{1}{\sqrt{d}}
  • B 1
  • C 0
  • D d\sqrt{d}

Step-by-Step Verified Solution

limn→∞dn(1a1+a2+1a2+a3+………+1an−1+an)

Now,

amp;1a1+a2+1a2+a3+…+1an−1+an=amp;a2−a1a2−a1+a3−a2a3−a2+…+an−an−1an−an−1=amp;a2−a1+a3−a2+..+an−an−1d\begin{aligned} & \frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}} \\\\ = & \frac{\sqrt{a_2}-\sqrt{a_1}}{a_2-a_1}+\frac{\sqrt{a_3}-\sqrt{a_2}}{a_3-a_2}+\ldots+\frac{\sqrt{a_n}-\sqrt{a_{n-1}}}{a_n-a_{n-1}} \\\\ = & \frac{\sqrt{a_2}-\sqrt{a_1}+\sqrt{a_3}-\sqrt{a_2}+. .+\sqrt{a_n}-\sqrt{a_{n-1}}}{d}\end{aligned}

(∵a2−a1=a3−a2=…an−an−1=d)\left(\because a_2-a_1=a_3-a_2=\ldots a_n-a_{n-1}=d\right)

amp;=an−a1damp;=a1+(n−1)d−a1d\begin{aligned} & =\frac{\sqrt{a_n}-\sqrt{a_1}}{d} \\\\ & =\frac{\sqrt{a_1+(n-1) d}-\sqrt{a_1}}{d}\end{aligned}

∴lim⁡n→∞dn(a1+(n−1)d−a1d)\therefore \lim\limits_{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left(\frac{\sqrt{a_1+(n-1) d}-\sqrt{a_1}}{d}\right)

amp;=lim⁡n→∞[1d(a1+(n−1)d−a1n)]amp;=lim⁡n→∞[1d(a1n+(d−dn)−a1n)]amp;=1d(0+d−0−0)=dd=1\begin{aligned} & =\lim _{n \rightarrow \infty}\left[\frac{1}{\sqrt{d}}\left(\frac{\sqrt{a_1+(n-1) d}-\sqrt{a_1}}{\sqrt{n}}\right)\right] \\\\ & =\lim _{n \rightarrow \infty}\left[\frac{1}{\sqrt{d}}\left(\sqrt{\frac{a_1}{n}+\left(d-\frac{d}{n}\right)}-\sqrt{\frac{a_1}{n}}\right)\right] \\\\ & =\frac{1}{\sqrt{d}}(\sqrt{0+d-0}-\sqrt{0})=\frac{\sqrt{d}}{\sqrt{d}}=1\end{aligned}

Related Learning Resources & Exam Preparation

JEE Main PYQ Question Bank
Browse year-wise authentic previous years' questions with solutions
Authentic CBT Mock Tests
Simulate official exam environment with live timer and percentile scoring