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

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

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

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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

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