Advertisements
Advertisements
प्रश्न
If n is a positive integer, find the coefficient of x–1 in the expansion of `(1 + x)^2 (1 + 1/x)^n`
उत्तर
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
संबंधित प्रश्न
Expand the expression (1– 2x)5
Expand the expression: (2x – 3)6
Expand the expression: `(x/3 + 1/x)^5`
Expand the expression: `(x + 1/x)^6`
Using binomial theorem, evaluate f the following:
(101)4
Find the coefficient of x5 in the product (1 + 2x)6 (1 – x)7 using binomial theorem.
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Show that \[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
Find the 4th term from the end in the expansion of `(x^3/2 - 2/x^2)^9`
Determine whether the expansion of `(x^2 - 2/x)^18` will contain a term containing x10?
Show that `2^(4n + 4) - 15n - 16`, where n ∈ N is divisible by 225.
Which of the following is larger? 9950 + 10050 or 10150
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 ______.
If z = `sqrt(3)/2 + i^5/2 + sqrt(3)/2 - i^5/2`, then ______.
Find the coefficient of x15 in the expansion of (x – x2)10.
If the coefficient of second, third and fourth terms in the expansion of (1 + x)2n are in A.P. Show that 2n2 – 9n + 7 = 0.
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 4OE = (x + a)2n – (x – a)2n
Given the integers r > 1, n > 2, and coefficients of (3r)th and (r + 2)nd terms in the binomial expansion of (1 + x)2n are equal, then ______.
The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1:4 are ______.
The coefficient of a–6b4 in the expansion of `(1/a - (2b)/3)^10` is ______.
Let the coefficients of x–1 and x–3 in the expansion of `(2x^(1/5) - 1/x^(1/5))^15`, x > 0, be m and n respectively. If r is a positive integer such that mn2 = 15Cr, 2r, then the value of r is equal to ______.
The sum of the last eight coefficients in the expansion of (1 + x)16 is equal to ______.
The positive integer just greater than (1 + 0.0001)10000 is ______.
