Advertisements
Advertisements
Question
Find the middle terms(s) in the expansion of:
(ii) \[\left( 1 - 2x + x^2 \right)^n\]
Advertisements
Solution
\[(1 - 2x + x^2 )^n \]
\[ = (1 - x )^{2n} \]
\[\text{ n is an even number } . \]
\[ \therefore \text{ Middle term } = \left( \frac{2n}{2} + 1 \right)th = (n + 1)\text{ th term} \]
\[\text{ Now, we have} \]
\[ T_{n + 1} = ^{2n}{}{C}_n ( - 1 )^n (x )^n \]
\[ = \frac{(2n)!}{(n! )^2}( - 1 )^n x^n\]
APPEARS IN
RELATED QUESTIONS
Write the general term in the expansion of (x2 – y)6
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`
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:
(iv) \[\left( \frac{x}{a} - \frac{a}{x} \right)^{10}\]
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:
(v) \[\left( x - \frac{1}{x} \right)^{2n + 1}\]
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:
(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) \[\left( \sqrt[3]{x} + \frac{1}{2 \sqrt[3]{x}} \right)^{18} , x > 0\]
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 2nd, 3rd and 4th terms in the expansion of (1 + x)2n are in A.P., show that \[2 n^2 - 9n + 7 = 0\]
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.
Find a, if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
If the coefficients of three consecutive terms in the expansion of (1 + x)n be 76, 95 and 76, find 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.
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.
Write the coefficient of the middle term in the expansion of \[\left( 1 + x \right)^{2n}\] .
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
In the expansion of \[\left( x^2 - \frac{1}{3x} \right)^9\] , the term without x is equal to
If in the expansion of \[\left( x^4 - \frac{1}{x^3} \right)^{15}\] , \[x^{- 17}\] occurs in rth term, then
If in the expansion of (1 + y)n, the coefficients of 5th, 6th and 7th terms are in A.P., then nis equal to
The total number of terms in the expansion of \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\] after simplification 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 the middle term (terms) in the expansion of `(p/x + x/p)^9`.
Find numerically the greatest term in the expansion of (2 + 3x)9, where x = `3/2`.
Find the middle term (terms) in the expansion of `(x/a - a/x)^10`
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 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 ______.
The number of terms in the expansion of [(2x + y3)4]7 is 8.
The last two digits of the numbers 3400 are 01.
If the expansion of `(x - 1/x^2)^(2n)` contains a term independent of x, then n is a multiple of 2.
The coefficient of y49 in (y – 1)(y – 3)(y – 5) ...... (y – 99) is ______.
