Careers360 Logo
Engineering and Architecture
Search Colleges, Exams, Schools & more
Best JEE Main Coaching Institutes in Bangalore with Fees Structure
  1. Home
  2. JEE Main
  3. Study Material
  4. Important Result (Comparison) - Practice Questions & MCQ

Important Result (Comparison) - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • 4 Questions around this concept.

Solve by difficulty

QuestionMEDIUM

If y=(x)^{3x} for x>2, which of the following is true?

View Solution

The value of the natural numbers n such that the inequality  2^n>2 n+1  is valid:

View Solution

Concepts Covered - 1

Important Result (Comparison)

Important Result (Comparison)

    2\leq\left ( 1+\frac{1}{n} \right )^n<3,\;\;\;\;\;n\in\mathbb{N}

Proof:

Expand, \\ \left ( 1+\frac{1}{n} \right )^n \text{using binomial theorem} \\\\

\begin{aligned}\left(1+\frac{1}{n}\right)^{n} &=1+n \frac{1}{n}+\frac{n(n-1)}{2 !} \frac{1}{n^{2}}+\frac{n(n-1)(n-2)}{3 !} \frac{1}{n^{3}}+\cdots+\frac{n(n-1)(n-2) \cdots[n-(n-1)]}{n !} \frac{1}{n^{n}} \\&={1+1+\frac{1}{2 !}\left(1-\frac{1}{n}\right)+\frac{1}{3 !}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)} {+\cdots+\frac{1}{n !}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right) \dots\left(1-\frac{n-1}{n}\right)}\\&{<1+1+\frac{1}{2 !}+\frac{1}{3 !}+\cdots+\frac{1}{n !}}\\&{<1+1+\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\cdots+\frac{1}{2^{n-1}}}=1+1 \frac{\left\{1-\left(\frac{1}{2}\right)^{n}\right\}}{1-\frac{1}{2}}=1+2\left\{1-\left(\frac{1}{2}\right)^{n}\right\}=3-\frac{1}{2^{n-1}}\end{aligned}

Hence, from above

2\leq\left ( 1+\frac{1}{n} \right )^n<3,\;\;n\geq1

Study it with Videos

Important Result (Comparison)

Study other Related Concepts

Important Result (Comparison) Current Topic

Binomial Theorem - Formula, Expansion, Problems and Applications
Some Standard Expansions
General and Middle Terms
Greatest Term (numerically)
An Important Theorem
Problems on Divisibility
Finding last digits
Important Result (Comparison)
Multinomial Theorem
Series Involving Binomial Coefficients

"Stay in the loop. Receive exam news, study resources, and expert advice!"

Books

Reference Books

Important Result (Comparison)

Mathematics Textbook for Class VII

Page No. : 8.4

Line : 14

Clear your Basics with NCERT

E-books & Sample Papers

Get Answer to all your questions

JEE Main Articles

UP BTech Admission 2024: Round 2 Choice Filling (Started), Seat Allotment, Merit List, Counselling, Process
Sep 07, 2024
Top Engineering Exams for NIOS Candidates
Sep 07, 2024
NIRF Ranking 2024 (Out) - List of Top Engineering Colleges in India
Sep 06, 2024
NIT Warangal Cutoff JEE Main 2024 - Check here
Sep 06, 2024
JEE Main 2025 Syllabus PDF - Subject-Wise Detailed Syllabus
Sep 06, 2024
Architecture Courses After 12th - Fees, Eligibility, Duration, Top Colleges
Sep 06, 2024
JEE Main Exam Pattern 2025 - Total Marks in JEE Mains, Paper Pattern, Duration
Sep 06, 2024
Top 10 Engineering Entrance Exams in India 2025
Sep 05, 2024

B.Tech/B.Arch Admissions OPEN

NIELIT University(Govt. of India Institution) B.Tech/ M.Tech
NIELIT University(Govt. of India Institution) B.Tech/ M.Tech
Apply

Last Date to Apply : 15th September | Campuses in Agartala, Aizawl, Ajmer, Aurangabad, Calicut, Imphal, Itanagar, Kohima, Gorakhpur, Patna & Srinagar

Amity University, Noida B.Tech Admissions 2024
Amity University, Noida B.Tech Admissions 2024
Apply

Asia's Only University with the Highest US & UK Accreditation

ITM SLS Baroda University B.Tech Admissions 2024
ITM SLS Baroda University B.Tech Admissions 2024
Apply

Approved by UGC, PCI, & COA. Best Innovative Private Institute in Gujarat. 600+ Recruiters, 3000+ Students Placed, 1500+ Industrial Internship Opportunities, 250+ International Internships.

MIT Academy of Engineering B.Tech Admissions 2024
MIT Academy of Engineering B.Tech Admissions 2024
Apply

Highest CTC 25 LPA | Average CTC 5.54 LPA | 280+ Recruiters

Plastindia International University B.Tech Admissions 2024
Plastindia International University B.Tech Admissions 2024
Apply

International Learning Opportunities | UGC Approved | State-of-the-Art Laboratories

Joy University B.Tech 2024
Joy University B.Tech 2024
Apply

Outstanding faculty of Eminence from IIT, BITS, IIM, IIIT NIT, NID and faculty from foreign universities | Merit & sports based scholarship

View All Application Forms
News and Notifications
JEE Main 2025 exam date this month; updates on reduced syllabus, registration, number of attempts
September 03, 2024 - 07:24 PM IST
AP BArch Admission 2024: Registration begins today at apsche.ap.gov.in; eligibility criteria
August 12, 2024 - 11:27 AM IST
JoSAA 2024 round 4 seat allotment results out at josaa.nic.in; report online by July 15
July 10, 2024 - 07:26 PM IST
CSAB NEUT 2024 registration begins at csab.nic.in; apply by June 28
June 20, 2024 - 12:07 PM IST
Youth preparing for IIT-JEE dies by suicide by hanging himself in Rajasthan's Kota
June 16, 2024 - 03:31 PM IST
JEE Main 2024: NTA debars 39 candidates for 3 years on account of using unfair means
May 02, 2024 - 10:31 AM IST
JEE Main 2024 session 2 paper 2 answer key objection window closes today; fees
May 01, 2024 - 08:44 AM IST

Download Careers360 App's

Regular exam updates, QnA, Predictors, College Applications & E-books now on your Mobile

  • student
    300M+

    Students

  • colleges
    36,000+

    Colleges

  • exams
    550+

    Exams

  • ebook
    1500+

    E-Books

  • certification
    16000+

    Certifications

student
Mobile Screen

We Appeared in

Joint Entrance Examination (Main)      
JEE Main 2025 Sample Papers
Back to top