Advertisements
Advertisements
प्रश्न
If xp occurs in the expansion of `(x^2 + 1/x)^(2n)`, prove that its coefficient is `(2n!)/(((4n - p)/3)!((2n + p)/3)!)`
उत्तर
Given expression is `(x^2 + 1/x)^(2n)`
General terms, `"T"_(r + 1) = ""^n"C"_rx^(n - r) y^r`
= `""^(2n)"C"_r (x^2)^(2n - r) * (1/x)^r`
= `""^(2n)"C"_r (x)^(4n - 2r) * 1/x^r`
= `""^(2n)"C"_r (x)^(4n - 2r - r)`
= `""^(2n)"C"_r(x)^(4n - 3r)`
If xp occurs in `(x^2 + 1/x)^(2n)`
Then 4n – 3r = p
⇒ 3r = 4n – p
⇒ r = `(4n - p)/3`
∴ Coefficient of xp = `""^(2n)"C"_r = ""^(""2n)"C"_((4n - p)/3)`
= `((2n)!)/(((4n - p)/3)!(2n - (4n - p)/3)!)`
= `((2n)!)/(((4n - p)/3)!((6n - 4n + p)/3)!)`
= `((2n)!)/(((4n - p)/3)!((2n + p)/3)!)`
Hence, the coefficient of xp = `((2n)!)/(((4n - p)/3)!((2n + p)/3)!)`
APPEARS IN
संबंधित प्रश्न
Find the coefficient of x5 in (x + 3)8
Write the general term in the expansion of (x2 – y)6
Find the middle terms in the expansions of `(x/3 + 9y)^10`
Find a positive value of m for which the coefficient of x2 in the expansion
(1 + x)m is 6
Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of `(root4 2 + 1/ root4 3)^n " is " sqrt6 : 1`
Find the middle terms in the expansion of:
(ii) \[\left( 2 x^2 - \frac{1}{x} \right)^7\]
Find the middle terms(s) in the expansion of:
(vi) \[\left( \frac{x}{3} + 9y \right)^{10}\]
Find the term independent of x in the expansion of the expression:
(iii) \[\left( 2 x^2 - \frac{3}{x^3} \right)^{25}\]
Find the term independent of x in the expansion of the expression:
(ix) \[\left( \sqrt[3]{x} + \frac{1}{2 \sqrt[3]{x}} \right)^{18} , x > 0\]
If the coefficients of \[\left( 2r + 4 \right)\text{ th and } \left( r - 2 \right)\] th terms in the expansion of \[\left( 1 + x \right)^{18}\] are equal, find r.
Prove that the term independent of x in the expansion of \[\left( x + \frac{1}{x} \right)^{2n}\] is \[\frac{1 \cdot 3 \cdot 5 . . . \left( 2n - 1 \right)}{n!} . 2^n .\]
Find a, if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
Find the coefficient of a4 in the product (1 + 2a)4 (2 − a)5 using binomial theorem.
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 p is a real number and if the middle term in the expansion of \[\left( \frac{p}{2} + 2 \right)^8\] is 1120, find p.
The number of irrational terms in the expansion of \[\left( 4^{1/5} + 7^{1/10} \right)^{45}\] is
The middle term in the expansion of \[\left( \frac{2x}{3} - \frac{3}{2 x^2} \right)^{2n}\] is
The number of terms with integral coefficients in the expansion of \[\left( {17}^{1/3} + {35}^{1/2} x \right)^{600}\] is
Find numerically the greatest term in the expansion of (2 + 3x)9, where x = `3/2`.
The ratio of the coefficient of x15 to the term independent of x in `x^2 + 2^15/x` is ______.
Find the term independent of x, x ≠ 0, in the expansion of `((3x^2)/2 - 1/(3x))^15`
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 coefficient of `1/x^17` in the expansion of `(x^4 - 1/x^3)^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 the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.
Middle term in the expansion of (a3 + ba)28 is ______.
The last two digits of the numbers 3400 are 01.
Let the coefficients of the middle terms in the expansion of `(1/sqrt(6) + βx)^4, (1 - 3βx)^2` and `(1 - β/2x)^6, β > 0`, common difference of this A.P., then `50 - (2d)/β^2` is equal to ______.
