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`
Advertisements
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: `(2/x - x/2)^5`
Expand the expression: `(x + 1/x)^6`
Using binomial theorem, evaluate f the following:
(101)4
Using binomial theorem, evaluate the following:
(99)5
Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.
Find (a + b)4 – (a – b)4. Hence, evaluate `(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
Find a, b and n in the expansion of (a + b)n if the first three terms of the expansion are 729, 7290 and 30375, respectively.
Find the value of `(a^2 + sqrt(a^2 - 1))^4 + (a^2 - sqrt(a^2 -1))^4`
Find an approximation of (0.99)5 using the first three terms of its expansion.
Find the value of (1.01)10 + (1 − 0.01)10 correct to 7 places of decimal.
Expand the following (1 – x + x2)4
Evaluate: `(x^2 - sqrt(1 - x^2))^4 + (x^2 + sqrt(1 - x^2))^4`
Find the coefficient of x11 in the expansion of `(x^3 - 2/x^2)^12`
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 .
The total number of terms in the expansion of (x + a)51 – (x – a)51 after simplification is ______.
If the coefficients of x7 and x8 in `2 + x^n/3` are equal, then n is ______.
The number of terms in the expansion of (a + b + c)n, where n ∈ N is ______.
Find the coefficient of x in the expansion of (1 – 3x + 7x2)(1 – x)16.
Find the coefficient of x15 in the expansion of (x – x2)10.
Find the sixth term of the expansion `(y^(1/2) + x^(1/3))^"n"`, if the binomial coefficient of the third term from the end is 45.
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.
Find the coefficient of x4 in the expansion of (1 + x + x2 + x3)11.
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
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 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 ______.
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 ______.
The positive integer just greater than (1 + 0.0001)10000 is ______.
