Home
Class 12
MATHS
Prove that ""^(n+1)C2+2*sum(k=2)^n""^kC2...

Prove that `""^(n+1)C_2+2*sum_(k=2)^n""^kC_2=sum_(k=1)^nk^2`

Text Solution

AI Generated Solution

Promotional Banner

Topper's Solved these Questions

  • Binomial Theorem for Positive Integrel Index

    A DAS GUPTA|Exercise Exercise|113 Videos

  • Application of dy/dx

    A DAS GUPTA|Exercise Exercise|45 Videos

  • Circles

    A DAS GUPTA|Exercise EXERCISE|122 Videos

Similar Questions

Explore conceptually related problems

For which positive integers n is the ratio (sum_(k=1)^(n)k^(2))/(sum_(k=1)^(n)k) an integer?

sum_(k=1)^(2n+1)(-1)^(k-1)k^(2)=

prove that sum_(k=1)^(n)k2^(-k)=2[1-2^(-n)-n*2^(-(n+1)))

If a_1,a_2,……….,a_(n+1) are in A.P. prove that sum_(k=0)^n ^nC_k.a_(k+1)=2^(n-1)(a_1+a_(n+1))

Prove that sum_(k=1)^(n-1) ""^(n)C_(k)[cos k x. cos (n+k)x+sin(n-k)x.sin(2n-k)x]=(2^(n)-2)cos nx .

For which positive integers n is the ratio, (sum+(k=1)^(n) k^(2))/(sum_(k=1)^(n) k) an integer ?

Let U_(n)=sum_(k=1)^(n)(n)/(n^(2)+k^(2)),S_(n)=sum_(k=0)^(n-1)(n)/(n^(2)+k^(2)) then

Fill in the blanks: (i) sum_(k=1)^(n)k=1+2+3+……………+n= ………… (ii) sum_(k=1)^(n)k^(2)=1^(2)+2^(2)+3^(2)+………………….+n^(2)= ……………….. (iii) sum_(k=1)^(n)k^(3)=1^(3)+2^(3)+3^(3)+…………..+n^(3)= …………………

A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. Evaluate (""^3C3+""^4C3+""^5C3+...+""^nC3)xx(""^nC3+""^nC4+""^nC5+...+...

    Text Solution

    |

  2. The value of ^n C1+^(n+1)C2+^(n+2)C3++^(n+m-1)Cm is equal to ^m+n C(n-...

    Text Solution

    |

  3. Prove that ""^(n+1)C2+2*sum(k=2)^n""^kC2=sum(k=1)^nk^2

    Text Solution

    |

  4. If (1+x)^n=C0+C1x+C2x^2+...+Cnx^n , find the sum of the following seri...

    Text Solution

    |

  5. Prove that ""^nC0+2*""^nC1+3*""^nC2+...+(n+1)""^nCn=(n+2)2^(n-1)

    Text Solution

    |

  6. Prove that ""^nC0+3*""^nC1+5*""^nC2+...+(2n+1)""^nCn=(n+1)2^(n)

    Text Solution

    |

  7. Prove that ""^nC0-2*""^nC1+3*""^nC2-...+(-1)""^n(n+1)""^nCn=0

    Text Solution

    |

  8. If sn=""^nC0+2*""^nC1+3*""^nC2+...+(n+1)*""^nCn then find sum(n=1)^oos...

    Text Solution

    |

  9. Find the sum :1*""^nC0+2*""^nC1+3*""^nC2+4*""^nC3+…, where n is an odd...

    Text Solution

    |

  10. Show that :""^nC0*m-""^nC1*(m-1)+""^nC2*(m-2)-...+(-1)^n*""^nCn*(m-n)=...

    Text Solution

    |

  11. Evaluate sum(r=1)^npr/r*"^nCr where pr denotes the sum of the first r ...

    Text Solution

    |

  12. Prove by binomial expansion that sum(k=1)^nk^2*"^nCk=n(n+1)2^(n-2)

    Text Solution

    |

  13. Evaluate sum(r=0)^n(r+1)^2*"^nCr

    Text Solution

    |

  14. If (1+x)^n=C0+C1x+C2x^2+C3x^3+...+Cnx^n then prove that 2.C0+2^2C1/2+2...

    Text Solution

    |

  15. If (1+x)^n=C0+C1x+C2x^2+C3x^3+...+Cnx^n then prove that C0-1/2C1+1/3C2...

    Text Solution

    |

  16. Prove that 3*""^10C0+3^2*(""^10C1)/2+3^3*(""^10C2)/3+...3^11*(""^10C10...

    Text Solution

    |

  17. Prove that 2*""^nC0+2^2*(""^nC1)/2+2^3*(""^nC2)/3+...2^(n+1)*(""^nCn)/...

    Text Solution

    |

  18. Find the sum sum(k=0)^n("^nCk)/((k+1)(k+2))

    Text Solution

    |

  19. Find the sum sum(r=0)^n(-1)^r*(""^nCr)/(""^(r+3)Cr)

    Text Solution

    |

  20. Find the sum sum(k=1) (""^nC(2k-1))/(2k)

    Text Solution

    |