Home
Class 12
MATHS
Prove that (^n C0)/x-(^n C0)/(x+1)+(^n C...

Prove that `(^n C_0)/x-(^n C_0)/(x+1)+(^n C_1)/(x+2)-+(-1)^n(^n C_n)/(x+n)=(n !)/(x(x+1)(x-n)),` where `n` is any positive integer and `x` is not a negative integer.

Text Solution

Verified by Experts

Let
`f(x) = (n!)/(x(x+1)(x+2)"...."(x+n))`
`= (A_(0))/(x) + (A_(1) )/(x+1) + (A_(2))/(x+2)+"...."+(A_(n))/(x+n)`.
(by partial fractions )
Then `A_(0)=[xf(x)]_(x=0)=(n!)/(1.2.3"...."n) =1=.^(n)C_(0)`
`A_(1) = [(x+1)f(x)]_(x) = - 1`
`= (n!)/((-1){1.2"....."(n-1)})`
`= (-(n!))/((n-1)!) = -.^(n)C_(1)`
`A_(2) = [(x+2)f(x)]_(x=-2)`
`=(n!)/((-2).(-1).1.2"....."(n-2))`
`= (n!)/(2!(n-2)!) = .^(n)C_(2)` and so on
Thus, `(n!)/(x(x+1)(x+2)"....."(x+n))`
`(.^(n)C_(0))/(x) - (.^(n)C_(1))/(x+1) + (.^(n)C_(2))/(x+2) - "....." + (-1)^(n) (.^(n)C_(n))/(x+n)" "(1)`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Concept Application Exercise 8.1|17 Videos

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Concept Application Exercise 8.2|10 Videos

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Matrix|4 Videos

  • AREA

    CENGAGE ENGLISH|Exercise Comprehension Type|2 Videos

  • CIRCLE

    CENGAGE ENGLISH|Exercise MATRIX MATCH TYPE|7 Videos

Similar Questions

Explore conceptually related problems

If n is a positive integer then int_(0)^(1)(ln x)^(n)dx is :

If f(x) =(p-x^n)^(1/n) , p >0 and n is a positive integer then f[f(x)] is equal to

Prove that 1-^n C_1(1+x)/(1+n x)+^n C_2(1+2x)/((1+n x)^2)-^n C_3(1+3x)/((1+n x)^3)+....(n+1) terms =0

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0) C_(n) - C_(1) C_(n-1) + C_(2) C_(n-2) - …+ (-1)^(n) C_(n) C_(0) = 0 or (-1)^(n//2) (n!)/((n//2)!(n//2)!) , according as n is odd or even .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0)^(2) - C_(1)^(2) + C_(2)^(2) -…+ (-1)^(n) *C_(n)^(2)= 0 or (-1)^(n//2) * (n!)/((n//2)! (n//2)!) , according as n is odd or even Also , evaluate C_(0)^(2) - C_(1)^(2) + C_(2)^(2) - ...+ (-1)^(n) *C_(n)^(2) for n = 10 and n= 11 .

If (1+x)^n=C_0+C_1x+C_2x^2+...+C_n x^n , t h e n C_0-(C_0+C_1)+(C_0+C_1+C_2)-(C_0+C_1+C_2+C_3)+ ...+(-1)^(n-1)(C_0+C_1+ C_(n-1)) , where n a) is even integer b) is a positive value c) a negative value d) divisible by 2^(n-1) divisible by 2^n

(d^n)/(dx^n)(logx)=? (a) ((n-1)!)/(x^n) (b) (n !)/(x^n) (c) ((n-2)!)/(x^n) (d) (-1)^(n-1)((n-1)!)/(x^n)

If 4^(x)*n^(2)=4^(x+1)*n and x and n are both positive integers, what is the value of n?

Prove that sum_(r=0)^n^n C_rsinr xcos(n-r)x=2^(n-1)sin(n x)dot

If f(x) = n , where n is an integer such that n le x lt n +1 , the range of f(x) is