Advertisements
Advertisements
प्रश्न
Write the coefficient of the middle term in the expansion of \[\left( 1 + x \right)^{2n}\] .
Advertisements
उत्तर
\[\text{ Here, n, i . e . , 2n, is an even number . } \]
\[ \therefore \text{ Middle term } = \left( \frac{2n}{2} + 1 \right)\text{ th term } = \left( n + 1 \right)\text{ th term}\]
\[\text{ Thus, we have } : \]
\[ T_{n + 1} =^{2n} C_n \left( 1 \right)^{2n - n} \left( x \right)^n \]
\[ =^{2n} C_n x^n \]
\[\text{ Hence, the coefficient of the middle term is } {}^{2n} C_n \]
APPEARS IN
संबंधित प्रश्न
Write the general term in the expansion of (x2 – yx)12, x ≠ 0
The coefficients of the (r – 1)th, rth and (r + 1)th terms in the expansion of (x + 1)n are in the ratio 1:3:5. Find n and r.
Prove that the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1 + x)2n–1 .
Find a positive value of m for which the coefficient of x2 in the expansion
(1 + x)m is 6
Find the middle term in the expansion of:
(iv) \[\left( \frac{x}{a} - \frac{a}{x} \right)^{10}\]
Find the middle terms in the expansion of:
(i) \[\left( 3x - \frac{x^3}{6} \right)^9\]
Find the middle terms in the expansion of:
(ii) \[\left( 2 x^2 - \frac{1}{x} \right)^7\]
Find the middle terms in the expansion of:
(iv) \[\left( x^4 - \frac{1}{x^3} \right)^{11}\]
Find the middle terms(s) in the expansion of:
(v) \[\left( x - \frac{1}{x} \right)^{2n + 1}\]
Find the middle terms(s) in the expansion of:
(vii) \[\left( 3 - \frac{x^3}{6} \right)^7\]
Find the middle terms(s) in the expansion of:
(viii) \[\left( 2ax - \frac{b}{x^2} \right)^{12}\]
Find the term independent of x in the expansion of the expression:
(ii) \[\left( 2x + \frac{1}{3 x^2} \right)^9\]
Find the term independent of x in the expansion of the expression:
(iv) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]
Find the term independent of x in the expansion of the expression:
(vi) \[\left( x - \frac{1}{x^2} \right)^{3n}\]
Find the term independent of x in the expansion of the expression:
(x) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^6\]
Prove that the term independent of x in the expansion of \[\left( x + \frac{1}{x} \right)^{2n}\] is \[\frac{1 \cdot 3 \cdot 5 . . . \left( 2n - 1 \right)}{n!} . 2^n .\]
If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)n are in A.P., then find the value of n.
If in the expansion of (1 + x)n, the coefficients of pth and qth terms are equal, prove that p + q = n + 2, where \[p \neq q\]
If in the expansion of (1 + x)n, the coefficients of three consecutive terms are 56, 70 and 56, then find n and the position of the terms of these coefficients.
If the 6th, 7th and 8th terms in the expansion of (x + a)n are respectively 112, 7 and 1/4, find x, a, n.
If the 2nd, 3rd and 4th terms in the expansion of (x + a)n are 240, 720 and 1080 respectively, find x, a, n.
Find the sum of the coefficients of two middle terms in the binomial expansion of \[\left( 1 + x \right)^{2n - 1}\]
If A and B are the sums of odd and even terms respectively in the expansion of (x + a)n, then (x + a)2n − (x − a)2n is equal to
The middle term in the expansion of \[\left( \frac{2 x^2}{3} + \frac{3}{2 x^2} \right)^{10}\] is
Find the middle term (terms) in the expansion of `(p/x + x/p)^9`.
Find numerically the greatest term in the expansion of (2 + 3x)9, where x = `3/2`.
If the term free from x in the expansion of `(sqrt(x) - k/x^2)^10` is 405, find the value of k.
Find the term independent of x in the expansion of `(3x - 2/x^2)^15`
Find n in the binomial `(root(3)(2) + 1/(root(3)(3)))^n` if the ratio of 7th term from the beginning to the 7th term from the end is `1/6`
If xp occurs in the expansion of `(x^2 + 1/x)^(2n)`, prove that its coefficient is `(2n!)/(((4n - p)/3)!((2n + p)/3)!)`
Find the term independent of x in the expansion of (1 + x + 2x3) `(3/2 x^2 - 1/(3x))^9`
If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.
In the expansion of `(x^2 - 1/x^2)^16`, the value of constant term is ______.
Middle term in the expansion of (a3 + ba)28 is ______.
The number of terms in the expansion of [(2x + y3)4]7 is 8.
The last two digits of the numbers 3400 are 01.
The coefficient of y49 in (y – 1)(y – 3)(y – 5) ...... (y – 99) is ______.
The term independent of x in the expansion of `[(x + 1)/(x^(2/3) - x^(1/3) + 1) - (x - 1)/(x - x^(1/2))]^10`, x ≠ 1 is equal to ______.
