Home
Class 12
MATHS
Find the coefficient of x^n in the polyn...

Find the coefficient of `x^n` in the polynomial `(x+^n C_0)(x+3^n C_1)xx(x+5^n C_2)[x+(2n+1)^n C_n]dot`

Text Solution

Verified by Experts

The correct Answer is:
`(n+1)2^(n)`
Given polynomial is `n +1` degree polynomial.
`(x+.^(n)C_(0))(x+3.^(n)C_(2))(x+5.^(n)C_(2))"....."[x+(2n+1).^(n)C_(n)]`
`= x^(n+1)+a_(1)x^(n)+"...."+a_(n+1)`
Then,
`- (a_(1))/(a_(0))=-.^(n)C_(0)-3.^(n)C_(1)-5.^(n)C_(2)-"....."-(2n+1).^(n)C_(n)`
`rArr a_(1)=.^(n)C_(0)+3.^(n)C_(1)+5.^(n)C_(2)-"......"+(2n+1).^(n)C_(n)`
`= underset(r=0)overset(n)sum.^(n)C_(r)(2r+1)=2underset(r=0)overset(n)sumr^(n)C_(r)+underset(r=0)overset(n)sum.^(n)C_(r)`
`= 2underset(r=0)overset(n)sumn..^(n-1)C_(r-1) + underset(r=0)overset(n)sum.^(n)C_(r)`
`= 2n 2^(n+1) + 2^(n) = (n+1)2^(n)`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Concept Application Exercise 8.7|9 Videos

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Concept Application Exercise 8.8|10 Videos

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Concept Application Exercise 8.5|8 Videos

  • AREA

    CENGAGE ENGLISH|Exercise Comprehension Type|2 Videos

  • CIRCLE

    CENGAGE ENGLISH|Exercise MATRIX MATCH TYPE|7 Videos

Similar Questions

Explore conceptually related problems

The coefficient of x^50 in the polynomial (x + ^50C_0)(x +3.^5C_1) (x +5.^5C_2).....(x + (2n + 1) ^5C_50) , is

The coefficient of x^(n) in the polynomial (x+""^(2n+1)C_(0))(X+""^(2n+1)C_(1)) (x+""^(2n+1)C_(2))……(X+""^(2n+1)C_(n)) is

If C_(0),C_(1), C_(2),...,C_(n) denote the cefficients in the expansion of (1 + x)^(n) , then C_(0) + 3 .C_(1) + 5 . C_(2)+ ...+ (2n + 1) C_(n) = .

If C_0,C_1,C_2..C_n denote the coefficients in the binomial expansion of (1 +x)^n , then C_0 + 2.C_1 +3.C_2+. (n+1) C_n

If n is not a multiple of 3, then the coefficient of x^n in the expansion of log_e (1+x+x^2) is : (A) 1/n (B) 2/n (C) -1/n (D) -2/n

The smallest natural number n, such that the coefficient of x in the expansion of (x^(2) + (1)/(x^(3)))^(n) is .^(n)C_(23) , is

Find the sum 3^n C_0-8^n C_1+13^n C_2xx^n C_3+dot

If C_(0), C_(1), C_(2),...,C_(n) denote the binomial coefficients in the expansion of (1 + x)^n) , then xC_(0)-(x -1) C_(1)+(x-2)C_(2)-(x -3)C_(3)+...+(-1)^(n) (x -n) C_(n)=

Prove that the coefficients of x^n in (1+x)^(2n) is twice the coefficient of x^n in (1+x)^(2n-1)dot

The coefficient of x^r[0lt=rlt=(n-1)] in the expansion of (x+3)^(n-1)+(x+3)^(n-2)(x+2)+(x+3)^(n-3)(x+2)^2+.... +(x+2)^(n-1) is a.^n C_r(3^r-2^n) b.^n C_r(3^(n-r)-2^(n-r)) c.^n C_r(3^r+2^(n-r)) d. none of these