Advertisements
Advertisements
प्रश्न
Find the middle term (terms) in the expansion of `(3x - x^3/6)^9`
Advertisements
उत्तर
Given expression is `(3x - x^3/6)^9`
Number of terms = 9 + 1 = 10 ....(even)
∴ Middle terms are `n^"th"/2` term and `(n/2 + 1)^"th"` term
= `10^"th"/2`
= 5th term and (5 + 1) = 6th term
General Term `"T"_(r + 1) = ""^n"C"_r x^(n - r) y^r`
∴ T5 = `"T"_(4 + 1)`
= `""^9"C"_4 (3x)^(9 - 4) (- x^3/6)^4`
= `""^9"C"_4 (3)^5 * x^5 (-1/6)^4 * x^12`
= `(9 xx 8 xx 7 xx 6)/(4 xx 3 xx 2 xx 1) xx (3 xx 3 xx 3 xx 3 xx 3)/(6 xx 6 xx 6 xx 6) x^17`
= `189/8 x^17`
Now, T6 = T5+1
= `""^9"C"_5 (3x)^(9 - 5) (- x^3/6)^5`
= `""^9"C"_5 (3)^4 x^4 (- 1/6)^5 * x^15`
= `(9 xx 8 xx 7 xx 6 xx 5)/(5 xx 4 xx 3 xx 2 xx 1) (3)^4 (- 1/6)^5 * x^19`
= ` - 21/16 x^19`
Hence, the required middle terms are `189/8 x^17` and `- 21/16 x^19`
APPEARS IN
संबंधित प्रश्न
Find the coefficient of x5 in (x + 3)8
Find the coefficient of a5b7 in (a – 2b)12
Find the 4th term in the expansion of (x – 2y)12 .
Find the middle terms in the expansions of `(3 - x^3/6)^7`
Find the middle terms in the expansions of `(x/3 + 9y)^10`
In the expansion of (1 + a)m + n, prove that coefficients of am and an are equal.
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 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 term in the expansion of:
(ii) \[\left( \frac{a}{x} + bx \right)^{12}\]
Find the middle terms(s) in the expansion of:
(ix) \[\left( \frac{p}{x} + \frac{x}{p} \right)^9\]
Find the term independent of x in the expansion of the expression:
(i) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9\]
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:
(vii) \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\]
If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)43 are equal, find r.
Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)n + 1 is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)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 a, b, c and d in any binomial expansion be the 6th, 7th, 8th and 9th terms respectively, then prove that \[\frac{b^2 - ac}{c^2 - bd} = \frac{4a}{3c}\].
If the 2nd, 3rd and 4th terms in the expansion of (x + a)n are 240, 720 and 1080 respectively, find x, a, n.
Write the middle term in the expansion of \[\left( x + \frac{1}{x} \right)^{10}\]
Write the coefficient of the middle term in the expansion of \[\left( 1 + x \right)^{2n}\] .
Write the total number of terms in the expansion of \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\] .
The middle term in the expansion of \[\left( \frac{2 x^2}{3} + \frac{3}{2 x^2} \right)^{10}\] is
In the expansion of \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\] , the term independent of x is
The number of terms with integral coefficients in the expansion of \[\left( {17}^{1/3} + {35}^{1/2} x \right)^{600}\] is
Find the middle term in the expansion of `(2ax - b/x^2)^12`.
The ratio of the coefficient of x15 to the term independent of x in `x^2 + 2^15/x` is ______.
Find the value of r, if the coefficients of (2r + 4)th and (r – 2)th terms in the expansion of (1 + x)18 are equal.
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`
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 ______.
The number of terms in the expansion of [(2x + y3)4]7 is 8.
The sum of coefficients of the two middle terms in the expansion of (1 + x)2n–1 is equal to 2n–1Cn.
The coefficient of x256 in the expansion of (1 – x)101(x2 + x + 1)100 is ______.
The sum of the co-efficients of all even degree terms in x in the expansion of `(x + sqrt(x^3 - 1))^6 + (x - sqrt(x^3 - 1))^6, (x > 1)` is equal to ______.
The middle term in the expansion of (1 – 3x + 3x2 – x3)6 is ______.
