Home
Class 11
MATHS
Prove that sum(r=0)^n^n Cr(-1)^r[i^r+i^(...

Prove that `sum_(r=0)^n^n C_r(-1)^r[i^r+i^(2r)+i^(3r)+i^(4r)]=2^n+2^(n/2+1)cos(npi//4),`
where `i=sqrt(-1)dot`

Text Solution

AI Generated Solution

Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS AND QUADRATIC EQUATIONS

    CENGAGE ENGLISH|Exercise All Questions|886 Videos

Similar Questions

Explore conceptually related problems

Prove that sum_(r=0)^n3^r^n C_r=4^n .

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

Prove that sum_(r = 0)^n r^2 . C_r = n (n +1).2^(n-2)

Prove that sum_(r = 0)^n r^3 . C_r = n^2 (n +3).2^(n-3)

Prove that sum_(r=0)^(2n)(r. ^(2n)C_r)^2=n^(4n)C_(2n) .

Prove that sum_(r=0)^(n) ""^(n)C_(r).(n-r)cos((2rpi)/(n)) = - n.2^(n-1).cos^(n)'(pi)/(n) .

Deduce that: sum_(r=0)^n .^nC_r (-1)^r 1/((r+1)(r+2)) = 1/(n+2)

Prove that sum_(r=0)^n r(n-r)(^nC_ r)^2=n^2(^(2n-2)C_n)dot

Prove that sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2.""^(4n-1)C_(2n-1) .

Prove that (3!)/(2(n+3))=sum_(r=0)^n(-1)^r((^n C_r)/(^(r+3)C_r))

CENGAGE ENGLISH-BINOMIAL THEOREM-All Questions
  1. If sum(r=0)^n{ar(x-alpha+2)^r-br(alpha-x-1)^r}=0, then prove that bn-(...

    Text Solution

    |

  2. Let a=(2^(1//401)-1) and for each ngeq2,l e tbn=^n C1+^n C2dota+^n C3...

    Text Solution

    |

  3. Prove that sum(r=0)^n^n Cr(-1)^r[i^r+i^(2r)+i^(3r)+i^(4r)]=2^n+2^(n/2+...

    Text Solution

    |

  4. Find the coefficient of x^n in (1+x/(1!)+(x^2)/(2!)+........+(x^n)/(n ...

    Text Solution

    |

  5. Prove that (^n C0)/x-(^n C0)/(x+1)+(^n C1)/(x+2)-+(-1)^n(^n Cn)/(x+n)=...

    Text Solution

    |

  6. If n is a positive integer, prove that 1-2n+(2n(2n-1))/(2!)-(2n(2n-1)(...

    Text Solution

    |

  7. Given, sn=1+q+q^2+.....+q^n ,Sn=1+(q+1)/2+((q+1)/2)^2+...+((q+1)/2)^n...

    Text Solution

    |

  8. The sum of 1+n(1-1/x)+(n(n+1))/(2!)(1-1/x)^2+oo will be a. x^n b. x^(-...

    Text Solution

    |

  9. sum(k=1)^ook(1-1/n)^(k-1)=>? a.n(n-1) b. n(n+1) c. n^2 d. (n+1)^2

    Text Solution

    |

  10. The coefficient of x^4 in the expansion of {sqrt(1+x^2)-x}^(-1) in asc...

    Text Solution

    |

  11. 1+1/3x+(1xx4)/(3xx6)x^2+(1xx4xx7)/(3xx6xx9)x^3+------ is equal to a. x...

    Text Solution

    |

  12. The value of sum(r=1)^(15)(r2^r)/((r+2)!) is (a).((17)!-2^16)/((17)!) ...

    Text Solution

    |

  13. (n+2)C0(2^(n+1))-(n+1)C1(2^(n))+(n)C2(2^(n-1))-.... is equal to

    Text Solution

    |

  14. The value of sum(r=0)^(50)(-1)^r((50)Cr)/(r+2) is equal to a . 1/(50x...

    Text Solution

    |

  15. In the expansion of [(1+x)//(1-x)]^2, the coefficient of x^n will be a...

    Text Solution

    |

  16. Statement : The sum of coefficient in the expansion of (3^(-x//4)+3^(5...

    Text Solution

    |

  17. Let n be a positive integer and k be a whole number, klt=2ndot Statem...

    Text Solution

    |

  18. Statement 1: sum sum(0le ilt j le n)(i/ (.^n ci)+j/(.^ncj)) is equal t...

    Text Solution

    |

  19. Statement 1: ^m Cr+^m C(r-1)(^nC1)+^mC(r-2)(^n C2)+....+^n Cr=0 , if m...

    Text Solution

    |

  20. 1+(1/4)+((1*3)/(4*8))+((1*4*7)/(4*8*12))+....=

    Text Solution

    |