Advertisements
Advertisements
Question
If n is a positive integer, find the coefficient of x–1 in the expansion of `(1 + x)^2 (1 + 1/x)^n`
Solution
We have `(1 + x)^n 1 + 1^n/x`
= `(1 + x)^n (x + 1)^n/x`
= `(1 + x)^(2n)/x^n`
Now to find the coefficient of x–1 in `(1 + x)^n 1 + 1^n/x`
It is equivalent to finding coefficient of x–1 in `(1 + x)^(2n)/x^n`
Which in turn is equal to the coefficient of xn–1 in the expansion of (1 + x)2n
Since (1 + x)2n = `""^(2n)"C"_0 x^0 + ""^(2n)"C"_1 + x^1 + ""^(2n)"C"_2 x^2 + ... + ""^(2n)"C"_(n - 1) x^(n - 1) + ... + ""^(2n)"C"_(2n) x^(2n)`
Thus the coefficient of `x^(n - 1)` is `""^(2n)"C"_(n - 1)`
= `(2n)/((n - 1)(2n - n + 1))`
= `(2n)/((n - 1)(n + 1))`
APPEARS IN
RELATED QUESTIONS
Expand the expression: (2x – 3)6
Expand the expression: `(x + 1/x)^6`
Using Binomial Theorem, evaluate the following:
(96)3
Using Binomial Theorem, evaluate of the following:
(102)5
Using binomial theorem, evaluate the following:
(99)5
Find (x + 1)6 + (x – 1)6. Hence or otherwise evaluate `(sqrt2 + 1)^6 + (sqrt2 -1)^6`
Prove that `sum_(r-0)^n 3^r ""^nC_r = 4^n`
Find a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
Find the coefficient of x5 in the product (1 + 2x)6 (1 – x)7 using binomial theorem.
Find an approximation of (0.99)5 using the first three terms of its expansion.
Find the expansion of (3x2 – 2ax + 3a2)3 using binomial theorem.
If n is a positive integer, prove that \[3^{3n} - 26n - 1\] is divisible by 676.
Find the rth term in the expansion of `(x + 1/x)^(2r)`
Expand the following (1 – x + x2)4
Evaluate: `(x^2 - sqrt(1 - x^2))^4 + (x^2 + sqrt(1 - x^2))^4`
Which of the following is larger? 9950 + 10050 or 10150
Find the coefficient of x50 after simplifying and collecting the like terms in the expansion of (1 + x)1000 + x(1 + x)999 + x2(1 + x)998 + ... + x1000 .
If a1, a2, a3 and a4 are the coefficient of any four consecutive terms in the expansion of (1 + x)n, prove that `(a_1)/(a_1 + a_2) + (a_3)/(a_3 + a_4) = (2a_2)/(a_2 + a_3)`
The total number of terms in the expansion of (x + a)51 – (x – a)51 after simplification is ______.
The coefficient of xp and xq (p and q are positive integers) in the expansion of (1 + x)p + q are ______.
In the expansion of (x + a)n if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that O2 – E2 = (x2 – a2)n
The total number of terms in the expansion of (x + a)100 + (x – a)100 after simplification is ______.
The sum of the last eight coefficients in the expansion of (1 + x)16 is equal to ______.
If the coefficients of (2r + 4)th, (r – 2)th terms in the expansion of (1 + x)18 are equal, then r is ______.
Let `(5 + 2sqrt(6))^n` = p + f where n∈N and p∈N and 0 < f < 1 then the value of f2 – f + pf – p is ______.
