Question:
The middle term in the expansion of $\left(1-\frac{1}{x}\right)^{n} \left(1-x^{n}\right)$ in powers of x is
The middle term in the expansion of $\left(1-\frac{1}{x}\right)^{n} \left(1-x^{n}\right)$ in powers of x is
Updated On: Jul 14, 2022
- $-^{2n}C_{n-1}$
- $-^{2n}C_{n}$
- $^{2n}C_{n-1}$
- $^{2n}C_{n}$
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Given expansion can be re-written as
$\left(1-\frac{1}{x}\right)^{n} .\left(1-x^{n}\right) = \left(-1\right)^{n}x^{-n} \left(1-x\right)^{2n}$
Total number of terms will be $2n + 1$ which is odd ($\because 2n$ is always even)
$\therefore$ Middle term $= \frac{2n+1+1}{2} = \left(n+1\right) \,th$
Now, $T_{r+1}=\,^{n}C_{r}\left(1\right)^{r} x^{n-r}$
So, $\frac{^{2n}C_{n} . x^{2n-n}}{x^{n}.\left(-1\right)^{n}} = ^{2n}C_{n} .\left(-1\right)^{n}$
Middle term is an odd term. So, $n + 1$ will be odd.
So, n will be even.
$\therefore$ Required answer is $^{2n}C_{n}.$
Was this answer helpful?
0
0
Top Questions on Binomial theorem
- The sum of the coefficients of \( x^{2/3} \) and \( x^{-2/5} \) in the binomial expansion of \[ \left( x^{2/3} + \frac{1}{2} x^{-2/5} \right)^9 \] is:
- JEE Main - 2024
- Mathematics
- Binomial theorem
- Let $m$ and $n$ be the coefficients of the seventh and thirteenth terms respectively in the expansion of \[\left( \frac{1}{3}x^{\frac{1}{3}} + \frac{1}{2x^{\frac{2}{3}}} \right)^{18}.\]Then \[\left(\frac{n}{m}\right)^{\frac{1}{3}}\]is:
- JEE Main - 2024
- Mathematics
- Binomial theorem
- If the term independent of \(x\) in the expansion of \[ \left( \sqrt{ax^2} + \frac{1}{2x^3} \right)^{10} \] is 105, then \(a^2\) is equal to:
- JEE Main - 2024
- Mathematics
- Binomial theorem
- Let $\alpha = \sum_{r=0}^n (4r^2 + 2r + 1) \binom{n}{r}$ and $\beta = \left( \sum_{r=0}^n \frac{\binom{n}{r}}{r+1} \right) + \frac{1}{n+1}$. If $140 < \frac{2\alpha}{\beta} < 281$, then the value of $n$ is _____.
- JEE Main - 2024
- Mathematics
- Binomial theorem
- Let $0 \leq r \leq n$. If \[^nC_{r+1} : ^nC_r : ^nC_{r-1} = 55 : 35 : 21,\]then $2n + 5r$ is equal to:
- JEE Main - 2024
- Mathematics
- Binomial theorem
View More Questions
Questions Asked in AIEEE exam
- This question has Statement 1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements. It is not possible to make a sphere of capacity $1$ farad using a conducting material. It is possible for earth as its radius is $6.4 \times 10^6\,m$.
- AIEEE - 2012
- electrostatic potential and capacitance
- A steel wire can sustain $100\, kg$ weight without breaking. If the wire is cut into two equal parts, each part can sustain a weight of
- AIEEE - 2012
- mechanical properties of solids
- The limiting line in Balmer series will have a frequency of (Rydberg constant, $R_{\infty}=3.29\times10^{15}$ cycles/s)
- AIEEE - 2012
- Electromagnetic Spectrum
- Which of the plots shown in the figure represents speed (v) of the electron in a hydrogen atom as a function of the principal quantum number (n)?
- The ratio of number of oxygen atoms (O) in 16.0 g ozone $(O_3), \,28.0\, g$ carbon monoxide $(CO)$ and $16.0$ oxygen $(O_2)$ is (Atomic mass: $C = 12,0 = 16$ and Avogadro?? constant $N_A = 6.0 x 10^{23}\, mol^{-1}$)
- AIEEE - 2012
- Mole concept and Molar Masses
View More Questions
Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.