Advertisements
Advertisements
प्रश्न
Find the middle terms(s) in the expansion of:
(vii) \[\left( 3 - \frac{x^3}{6} \right)^7\]
उत्तर
\[\left( 3 - \frac{x^3}{6} \right)^7 \]
\[\text{ Here, n is an odd number } . \]
\[\text{ Therefore, the middle terms are } \left( \frac{7 + 1}{2} \right)\text{ th and } \left( \frac{7 + 1}{2} + 1 \right)\text{ th, i . e . , 4th and 5th terms . } \]
\[\text{ Now, we have } \]
\[ T_4 = T_{3 + 1} \]
\[ =^{7}{}{C}_3 \left( 3 \right)^{7 - 3} \left( \frac{- x^3}{6} \right)^3 \]
\[ = - \frac{105}{8} x^9 \]
\[\text{ And} , \]
\[ T_5 = T_{4 + 1} \]
\[ = ^{9}{}{C}_4 \left( 3 \right)^{9 - 4} \left( \frac{- x^3}{6} \right)^4 \]
\[ = \frac{7 \times 6 \times 5}{3 \times 2} \times 3^5 \times \frac{1}{6^4} x^{12} \]
\[ = \frac{35}{48} x^{12}\]
APPEARS IN
संबंधित प्रश्न
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.
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 terms in the expansion of:
(ii) \[\left( 2 x^2 - \frac{1}{x} \right)^7\]
Find the middle terms in the expansion of:
(iii) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]
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:
(iv) \[\left( 2x - \frac{x^2}{4} \right)^9\]
Find the middle terms(s) in the expansion of:
(x) \[\left( \frac{x}{a} - \frac{a}{x} \right)^{10}\]
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:
(vi) \[\left( x - \frac{1}{x^2} \right)^{3n}\]
Find the term independent of x in the expansion of the expression:
(vii) \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\]
Find the term independent of x in the expansion of the expression:
(ix) \[\left( \sqrt[3]{x} + \frac{1}{2 \sqrt[3]{x}} \right)^{18} , x > 0\]
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 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.
The coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)n are in A.P., find 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 \[p \neq q\]
Find the coefficient of a4 in the product (1 + 2a)4 (2 − a)5 using binomial theorem.
In the expansion of (1 + x)n the binomial coefficients of three consecutive terms are respectively 220, 495 and 792, find the value of n.
If 3rd, 4th 5th and 6th terms in the expansion of (x + a)n be respectively a, b, c and d, prove that `(b^2 - ac)/(c^2 - bd) = (5a)/(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.
If the term free from x in the expansion of \[\left( \sqrt{x} - \frac{k}{x^2} \right)^{10}\] is 405, find the value of k.
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.
Write the middle term in the expansion of `((2x^2)/3 + 3/(2x)^2)^10`.
If the sum of odd numbered terms and the sum of even numbered terms in the expansion of \[\left( x + a \right)^n\] are A and B respectively, then the value of \[\left( x^2 - a^2 \right)^n\] is
The middle term in the expansion of \[\left( \frac{2x}{3} - \frac{3}{2 x^2} \right)^{2n}\] is
If rth term is the middle term in the expansion of \[\left( x^2 - \frac{1}{2x} \right)^{20}\] then \[\left( r + 3 \right)^{th}\] term 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`.
Find numerically the greatest term in the expansion of (2 + 3x)9, where x = `3/2`.
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.
If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.
The sum of coefficients of the two middle terms in the expansion of (1 + x)2n–1 is equal to 2n–1Cn.
If n is the number of irrational terms in the expansion of `(3^(1/4) + 5^(1/8))^60`, then (n – 1) is divisible by ______.
The middle term in the expansion of (1 – 3x + 3x2 – x3)6 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 ______.
