NIT JEE Main Cutoff 2025 - NIT BTech Admission Process

Important Results of Binomial Theorem for any Index - Practice Questions & MCQ

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

Quick Facts

Important Results of Binomial Theorem for any Index is considered one the most difficult concept.
47 Questions around this concept.

Solve by difficulty

If the expansion in powers of $x$ of the function $\dpi{100} \frac{1}{(1-ax)(1-bx)}\,\,is\, \,\,a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+.......,\, then\; a_{n}\; is$ :

A

$\frac{b^{n}-a^{n}}{b-a}\;$

B

$\; \; \frac{a^{n}-b^{n}}{b-a}\; \;$

C

$\; \frac{a^{n+1}-b^{n+1}}{b-a}\; \;$

D

$\; \; \frac{b^{n+1}-a^{n+1}}{b-a}\; \; \;$

If 0 < $x$ < 1, then the first negative term in the expansion of $\dpi{100} (1+x)^{41/7}$ is:

A

5^th term

B

8^th term

C

6^th term

D

7^th term

Concepts Covered - 1

Important Results of Binomial Theorem for any Index

Important Results of Binomial Theorem for any Index

$\\\text{If the given series is}\\\\\mathrm{(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+......+\frac{n(n-1)(n-2).....(n-r+1)}{r!}x^r....}$

In the above expansion replace ‘n’ with ‘-n’

$\\\\\mathrm{(1+x)^{-n}=1+(-n)x+\frac{(-n)((-n)-1)}{2!}x^2+\frac{(-n)((-n)-1)((-n)-2)}{3!}x^3+......}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;....+\frac{(-n)((-n)-1)((-n)-2).....((-n)-r+1)}{r!}x^r.........\infty}$

$\Rightarrow \mathrm{(1+x)^{-n}=1-nx+\frac{n(n+1)}{2!}x^2-\frac{n(n+1)(n+2)}{3!}x^3+......}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;....+(-1)^r\frac{n(n+1)(n+2).....(n+r-1)}{r!}x^r.........\infty}$

If -n is a negative integer (so that n is a positive integer), then we can re-write this expression as

$=1-^{n} C_{1} x+^{n+1} C_{2} x^{2}-^{n+2} C_{3} x^{3}+\cdots+^{n+r-1} C_{r}(-x)^{r}+\cdots$

Now replace ‘x’ with ‘-x’ and ‘n’ with ‘-n’ in the binomial expansion (1 + x)ⁿ.

$\\(1-x)^{-n}=1+ n x+\frac{n(n+1)}{2 !} x^{2}+\frac{n(n+1)(n+2)}{3 !} x^{3}+\cdots \\ +\frac{n(n+1)(n+2) \cdots(n+r-1)}{r !} x^{r}+\cdots\\\\\text{If -n is a negative integer (so that n is a positive integer), then we can re-write this expression as } \\\\=1+^{n} C_{1} x+^{n+1} C_{2} x^{2}+^{n+2} C_{3} x^{3}+\cdots+^{n+r-1} C_{r}(x)^{r}+\cdots$

Important Note:

The coefficient of x^r in (1 - x)^-n, (when n is a natural number) is ^n+r-1C_r

Some Important Binomial Expansion

$\\1.\;\;(1+x)^{-1}=1-x+x^{2}-x^{3}+\cdots\\2.\;\;(1-x)^{-1}=1+x+x^{2}+x^{3}+\cdots\\\3.\;\;(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots\\4.\;\;(1-x)^{-2}=1+2 x+3 x^{2}+4 x^{3}+\cdots$

Study it with Videos

Important Results of Binomial Theorem for any Index

Study other Related Concepts

Important Results of Binomial Theorem for any Index 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!"

Get Answer to all your questions

JEE Main Articles

JEE Main Important Formulas 2025 for Physics, Chemistry, Maths

Nov 21, 2024

Scholarship for BTech Students in India 2025 - Eligibility, Amount, How to Apply

Nov 21, 2024

JEE Main Registration 2025 Last Date Tomorrow, Direct Link at jeemain.nta.nic.in, Live Updates

Nov 21, 2024

JEE Main 2025 Exam Centres in Andhra Pradesh (AP)

Nov 21, 2024

Upcoming Engineering Exams 2025 - Latest Updates on JEE, VITEEE, BITSAT, MHT CET, State Exams

Nov 21, 2024

Engineering Entrance Exams Application Date 2025 & Details

Nov 21, 2024

MNNIT Allahabad Cutoff JEE Main 2024 - Check Previous Year Cutoff Here

Nov 21, 2024

NIT Calicut Cutoff JEE Main 2024 - Opening and Closing Rank

Nov 21, 2024

B.Tech/B.Arch Admissions OPEN

UPES B.Tech Admissions 2025

Ranked #42 among Engineering colleges in India by NIRF | Highest CTC 50 LPA , 100% Placements

Amrita University B.Tech 2025

Recognized as Institute of Eminence by Govt. of India | NAAC ‘A++’ Grade | Upto 75% Scholarships

Dayananda Sagar University B.Tech Admissions 2025

60+ Years of Education Legacy | UGC & AICTE Approved | Prestigious Scholarship Worth 6 Crores | H-CTC 35 LPA

Parul University B.Tech Admissions 2025

India's youngest NAAC A++ accredited University | NIRF rank band 151-200 | 2200 Recruiters | 45.98 Lakhs Highest Package

Lovely Professional University B.Tech Admissions 2025

India's Largest University | NAAC A++ Accredited | 100% Placement Record | Highest Package Offered : 3 Cr PA

Atlas SkillTech University | B.Tech Admissions

Hands on Mentoring and Code Coaching | Cutting Edge Curriculum with Real World Application

View All Application Forms

News and Notifications

12 hours ago :

Check JEE Main 2025 registration live updates here.

BTech Admission 2025 Live: JEE Mains registrations closing tomorrow; VITEEE, MET exam dates out

November 21, 2024 - 03:28 PM IST

JEE Main 2025 registration ends tomorrow; changes introduced in exam this year

November 21, 2024 - 08:34 AM IST

JEE Main 2025 Live: NTA to close JEE registration on Friday; apply at jeemain.nta.nic.in, correction window

November 20, 2024 - 09:45 PM IST

BTech Exams 2025: JEE Main application correction dates out; MHT CET syllabus, VITEEE exam date

November 19, 2024 - 10:05 PM IST

JEE Mains 2025 session 1 registration closing soon; application correction window open till November 27

November 19, 2024 - 03:45 PM IST

JEE Main 2025 exam date will not be changed in case of clash with other exams; NTA answers FAQs

November 19, 2024 - 07:59 AM IST

JEE Mains 2025: Application correction facility from November 26 to 27; NTA answers FAQs

November 18, 2024 - 10:52 PM IST

Download Careers360 App

All this at the convenience of your phone

Regular Exam Updates
Best College Recommendations
College & Rank predictors
Detailed Books and Sample Papers
Question and Answers

Scan and download the app

OR

Joint Entrance Examination (Main)

JEE Main 2025 Sample Papers