Advertisements
Advertisements
प्रश्न
Find the middle terms in the expansion of:
(i) \[\left( 3x - \frac{x^3}{6} \right)^9\]
Advertisements
उत्तर
(i) Here, n, i.e. 9, is an odd number.
Thus, the middle terms are \[\left( \frac{n + 1}{2} \right)\text{th and } \left( \frac{n + 1}{2} + 1 \right)th, \text{ i . e . 5th and 6th } \]
\[Now, \]
\[ T_5 = T_{4 + 1} = ^{9}{}{C}_4 (3x )^{9 - 4} \left( \frac{- x^3}{6} \right)^4 \]
\[ = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2} \times 27 \times 9 \times \frac{1}{36 \times 36} x^{17} \]
\[ = \frac{189}{8} x^{17} \]
\[\text{ and } , \]
\[ T_6 = T_{5 + 1} \]
\[ =^{9}{}{C}_5 (3x )^{9 - 5} \left( \frac{- x^3}{6} \right)^5 \]
\[ = - \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2} \times 81 \times \frac{1}{216 \times 36} x^{19} \]
\[ = - \frac{21}{16} x^{19}\]
APPEARS IN
संबंधित प्रश्न
Find the 13th term in the expansion of `(9x - 1/(3sqrtx))^18 , 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.
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:
(iii) \[\left( x^2 - \frac{2}{x} \right)^{10}\]
Find the middle terms(s) in the expansion of:
(i) \[\left( x - \frac{1}{x} \right)^{10}\]
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:
(viii) \[\left( 2ax - \frac{b}{x^2} \right)^{12}\]
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:
(ii) \[\left( 2x + \frac{1}{3 x^2} \right)^9\]
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:
(vi) \[\left( x - \frac{1}{x^2} \right)^{3n}\]
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.
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.
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 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 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.
If in the expansion of (a + b)n and (a + b)n + 3, the ratio of the coefficients of second and third terms, and third and fourth terms respectively are equal, then n is
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
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
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`
Find the middle term (terms) in the expansion of `(3x - x^3/6)^9`
Find the coefficient of `1/x^17` in the expansion of `(x^4 - 1/x^3)^15`
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.
Show that the middle term in the expansion of `(x - 1/x)^(2x)` is `(1 xx 3 xx 5 xx ... (2n - 1))/(n!) xx (-2)^n`
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 expansion of `(x - 1/x^2)^(2n)` contains a term independent of x, then n is a multiple of 2.
