Home
Class 11
MATHS
If k a n d n are positive integers and s...

If `k a n d n` are positive integers and `s_k=1^k+2^k+3^k++n^k ,` then prove that `sum_(r=1)^m^(m+1)C_r s_r=(n+1)^(m+1)-(n+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

Show that sum_(k=m)^n ^kC_r=^(n+1)C_(r+1)-^mC_(r+1)

If n is a positive integer and C_(k)=""^(n)C_(k) , then the value of sum_(k=1)^(n)k^(3)((C_(k))/(C_(k-1)))^(2) is :

sum_(k=m)^n kC_r

Let k=1^@ , then prove that sum_(n=0)^88 1/(cosnk* cos(n+1)k)=cosk/sin^2k

Find the sum_(k=1)^(oo) sum_(n=1)^(oo)k/(2^(n+k)) .

If n and k are positive integers, show that 2^k( .^n C_0)(.^n C_k)-2^(k-1)(.^n C_1)(.^(n-1) C_k-1)+2^(k-2)(.^n C_2)((n-2k-2))_dot-...+ (-1)^k(^n C_k)+(.^(n-k) C_0)=(.^n C_k)w h e r e(.^n C_k) stands for .^n C_k.

If m,n,r are positive integers such that r lt m,n, then ""^(m)C_(r)+""^(m)C_(r-1)""^(n)C_(1)+""^(m)C_(r-2)""^(n)C_(2)+...+ ""^(m)C_(1)""^(n)C_(r-1)+""^(n)C_(r) equals

Prove that sum_(k=0)^(n) (-1)^(k).""^(3n)C_(k) = (-1)^(n). ""^(3n-1)C_(n)

Evaluate : sum_(k=1)^n (2^k+3^(k-1))

If (1+x)^n=sum_(r=0)^n C_r x^r , then prove that C_1+2C_2+3C_3+....+n C_n=n2^(n-1)dot .

CENGAGE ENGLISH-BINOMIAL THEOREM-All Questions
  1. Find the sum sum(i=0)^r.^(n1)C(r-i) .^(n2)Ci .

    Text Solution

    |

  2. Prove that sum(r=0)^(2n)(r. ^(2n)Cr)^2=n^(4n)C(2n) .

    Text Solution

    |

  3. If k a n d n are positive integers and sk=1^k+2^k+3^k++n^k , then prov...

    Text Solution

    |

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

    Text Solution

    |

  5. Prove that (C1)/1-(C2)/2+(C3)/3-(C4)/4++((-1)^(n-1))/n Cn=1+1/2+1/3++1...

    Text Solution

    |

  6. Prove that sum(r=0)^n^n Crsinr xcos(n-r)x=2^(n-1)sin(n x)dot

    Text Solution

    |

  7. Find the last two digits of the number (23)^(14)dot

    Text Solution

    |

  8. Find the last two digits of the number 27^(27)dot

    Text Solution

    |

  9. Find the number of nonzero terms in the expansion of (1+3sqrt(2)x)^9+(...

    Text Solution

    |

  10. Find the value of (sqrt(2)+1)^6-(sqrt(2)-1)^6dot

    Text Solution

    |

  11. Using binomial theorem (without using the formula for .^n Cr) , prove ...

    Text Solution

    |

  12. Prove that .(r+1)*^n Cr-r*^n Cr+ ... +(-1)^r.^n Cr=(-1)^r.^(n-2)Crdot

    Text Solution

    |

  13. Find the sum .^n C0+^n C4+^n C8 + . . .

    Text Solution

    |

  14. Find the value of ^4nC0+^(4n)C4+^(4n)C8++""^(4n)C(4n) .

    Text Solution

    |

  15. Find the coefficient of x^n in the polynomial (x+^n C0)(x+3^n C1)xx(x+...

    Text Solution

    |

  16. If (1+x)^(15)=C0+C1x+C2x^2++C(15)x^(15), then find the value of C2+2C...

    Text Solution

    |

  17. Prove that (.^n C0)/1+(.^n C2)/3+(.^n C4)/5+(.^n C6)/7+ . . . =(2^n)/(...

    Text Solution

    |

  18. Find the sum sumsum(0lt=ilt=jlt=n)^n Ci^n Cj

    Text Solution

    |

  19. Find the sum sumsum(i!=j)^n ^nCi^n Cj

    Text Solution

    |

  20. Show that the integer next above (sqrt(3)+1)^(2m) contains 2^(m+1) as ...

    Text Solution

    |