Advertisements
Advertisements
Question
Find the middle terms(s) in the expansion of:
(vi) \[\left( \frac{x}{3} + 9y \right)^{10}\]
Advertisements
Solution
\[\left( \frac{x}{3} + 9y \right)^{10} \]
\[\text{ Here, n is an even number } . \]
\[\text{ Therefore, the middle term is } \left( \frac{10}{2} + 1 \right)\text{ th, i . e . , 6th term . } \]
\[\text{ Now, we have } \]
\[ T_6 = T_{5 + 1} \]
\[ =^{10}{}{C}_5 \left( \frac{x}{3} \right)^{10 - 5} (9y )^5 \]
\[ = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2} \times \frac{1}{3^5} \times 9^5 \times x^5 y^5 \]
\[ = 61236 x^5 y^5\]
APPEARS IN
RELATED QUESTIONS
Find the coefficient of x5 in (x + 3)8
Write the general term in the expansion of (x2 – y)6
Write the general term in the expansion of (x2 – yx)12, x ≠ 0
Find the middle terms in the expansions of `(3 - x^3/6)^7`
Find the middle term in the expansion of:
(iv) \[\left( \frac{x}{a} - \frac{a}{x} \right)^{10}\]
Find the middle terms(s) in the expansion of:
(ii) \[\left( 1 - 2x + x^2 \right)^n\]
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:
(vii) \[\left( 3 - \frac{x^3}{6} \right)^7\]
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:
(v) \[\left( \frac{\sqrt{x}}{3} + \frac{3}{2 x^2} \right)^{10}\]
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 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 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 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.
Find a, b and n in the expansion of (a + b)n, if the first three terms in the expansion are 729, 7290 and 30375 respectively.
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 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
If an the expansion of \[\left( 1 + x \right)^{15}\] , the coefficients of \[\left( 2r + 3 \right)^{th}\text{ and } \left( r - 1 \right)^{th}\] terms are equal, then the value of r is
The middle term in the expansion of \[\left( \frac{2x}{3} - \frac{3}{2 x^2} \right)^{2n}\] is
Find the middle term in the expansion of `(2ax - b/x^2)^12`.
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 the term independent of x in the expansion of (1 + x + 2x3) `(3/2 x^2 - 1/(3x))^9`
Middle term in the expansion of (a3 + ba)28 is ______.
The sum of coefficients of the two middle terms in the expansion of (1 + x)2n–1 is equal to 2n–1Cn.
The number of rational terms in the binomial expansion of `(4^(1/4) + 5^(1/6))^120` 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 ______.
Let for the 9th term in the binomial expansion of (3 + 6x)n, in the increasing powers of 6x, to be the greatest for x = `3/2`, the least value of n is n0. If k is the ratio of the coefficient of x6 to the coefficient of x3, then k + n0 is equal to ______.
