Home
Class 12
MATHS
If (1+x)^n=C0+C1\ x+C2\ x^2+\ +Cn\ x^n ,...

If `(1+x)^n=C_0+C_1\ x+C_2\ x^2+\ +C_n\ x^n` , using derivatives prove that
(I)`C_1+2C_2+\ dot+n C_n=n .2^(n-1)`
(ii) `C_1-2C_2+3C_3+\ dot+(-1)^(n-1)\ n C_n=0`

Text Solution

Verified by Experts

Given that,
`(1+x)^n=C_0+C_1\ x+C_2\ x^2+\ +C_n\ x^n`

(i) `(1+x)^n=C_0+C_1\ x+C_2\ x^2+\ +C_n\ x^n`

Differentiating w.r.t `x` both sides

`n(1+x)^(n-1)=0+C_1+2C_2\ x+\ \...+nC_n\ x^(n-1)`

Putting `x=1`

`n(1+1)^(n-1)=0+C_1+2C_2\ (1)+\ \...+nC_n\ (1)^(n-1)`

`n(2)^(n-1)=C_1+2C_2+\ \...+nC_n\ `


...
Promotional Banner

Topper's Solved these Questions

  • DIFFERENTIALS, ERRORS AND APPROXIMATIONS

    RD SHARMA|Exercise Solved Examples And Exercises|102 Videos

  • DIRECTION COSINES AND DIRECTION RATIOS

    RD SHARMA|Exercise Solved Examples And Exercises|67 Videos

Similar Questions

Explore conceptually related problems

If (1+x)^(n)=c_(0)+C_(1)x+C_(2)x^(2)++C_(n)x^(n) ,using derivative prove that C_(1)+2C_(2)++nC_(n)=n.2^(n-1)C_(1)-2C_(2)+3C_(3)++(-1)^(n-1)nC_(n)=0

If (1+x)^n = C_0 + C_1x + C_2x^2 + ………. + C_n x^n , prove that : C_0 + 2C_1 + ….. + 2 ""^nC_n = 3^n

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+....+C_(n)x^(n) , then prove that : (i) C_(1)+2C_(2)+......+nC_(n)=n*2^(n-1) (ii) C_(1)-2C_(2)+......+(-1)^(n-1)nC_(n)=0

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) +… + C_(n) x^(n) , prove that C_(0) + 2C_(1) + 3C_(2) + …+ (n+1)C_(n) = (n+2)2^(n-1) .

If (1+ x)^(n) = C_(0) + C_(1) x + C_(2)x^(2) + ...+ C_(n)x^(n) , prove that C_(1) + 2C_(2) + 3C_(3) + ...+ n""C_(n) = n*2^(n-1)

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_2x^2 + …………+C_n x^n , find the value of C_0 - 2C_1 + 3C_2 - ……….+ (-1)^n (n+1) C_n

If (1+x)^n = C_0 + C_1x + C_2x^2 + ………. + C_n x^n , prove that : C_0 + (C_1)/(2) + (C_2)/(3) + ……. + (C_n)/(n+1) = (2^(n+1) -1)/(n+1)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n)," prove that " + 3^(2) *C_(3) + …+ n^(2) *C_(n) 1^(2)*C_(1) + 2^(2) *C_(2) = n(n+1)* 2^(n-2) .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(0) x^(n) , prove that (1*2) C_(2) + (2*3) C_(3) + …+ {(n-1)*n} C_(n) = n(n-1) 2^(n-2) .