Home
Class 12
MATHS
If a(1),a(2),a(3),…a(n+1) are in arithme...

If `a_(1),a_(2),a_(3),…a_(n+1)` are in arithmetic progression, then `sum_(k=0)^(n) .^(n)C_(k.a_(k+1)` is equal to (a)`2^(n)(a_(1)+a_(n+1))` (b)`2^(n-1)(a_(1)+a_(n+1))` (c)`2^(n+1)(a_(1)+a_(n+1))` (d)`(a_(1)+a_(n+1))`

A

`2^(n)(a_(1)+a_(n+1))`

B

`2^(n-1)(a_(1)+a_(n+1))`

C

`2^(n+1)(a_(1)+a_(n+1))`

D

`(a_(1)+a_(n+1))`

Text Solution

Verified by Experts

The correct Answer is:
B
`sum_(k=0)^(n) .^(n)C_(k) a_(k-1)=sum_(k=0)^(n) .^(n)C_(k)(a_(1)+kd)`
`=a_(1)sum_(k=0)^(n) .^(n)C_(k)+dn=sum_(k=0)^(n) .^(n-1)C_(k-1)`
`=8_(1)2^(n)+nd.2^(n-1)`
`=2^(n-1)(2B_(1)+nd)`
`=2^(n-1)(a_(n+1)+a_(1))`
Promotional Banner

Topper's Solved these Questions

  • TEST PAPERS

    RESONANCE ENGLISH|Exercise Math|105 Videos

  • TEST PAPER

    RESONANCE ENGLISH|Exercise MATHEMATICS|48 Videos

  • TEST SERIES

    RESONANCE ENGLISH|Exercise MATHEMATICS|132 Videos

Similar Questions

Explore conceptually related problems

If A_(1), A_(2),..,A_(n) are any n events, then

If (a_(1)+ib_(1))(a_(2)+ib_(2))………………(a_(n)+ib_(n))=A+iB , then sum_(i=1)^(n) tan^(-1)(b_(i)/a_(i)) is equal to

If a_(1),a_(2),a_(3),".....",a_(n) are in HP, than prove that a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+"....."+a_(n-1)a_(n)=(n-1)a_(1)a_(n)

If a_1,a_2,a_3,... are in A.P. and a_i>0 for each i, then sum_(i=1)^n n/(a_(i+1)^(2/3)+a_(i+1)^(1/3)a_i^(1/3)+a_i^(2/3)) is equal to (a) (n)/(a_(n)^(2//3)+a_(n)^(1//3)+a_(1)^(2//3)) (b) (n(n+1))/(a_(n)^(2//3)+a_(n)^(1//3)+a_(1)^(2//3)) (c) (n(n-1))/(a_(n)^(2//3)+a_(n)^(1//3)*a_(1)^(1//3)+a_(1)^(2//3)) (d) None of these

If the sequence a_(1),a_(2),a_(3),…,a_(n) is an A.P., then prove that a_(1)^(2)-a_(2)^(2)+a_(3)^(2)-a_(4)^(2)+…+a_(2n-1)^(2)-a_(2n)^(2)=n/(2n-1)(a_(1)^(2)-a_(2n)^(2))

If a_(1)=3 and a_(n)=2a_(n-1)+5 , find a_(4) .

If a_(1),a_(2),a_(3),"........",a_(n) are in AP with a_(1)=0 , prove that (a_(3))/(a_(2))+(a_(4))/(a_(3))+"......"+(a_(n))/(a_(n-1))-a_(2)((1)/(a_(2))+(1)/(a_(3))"+........"+(1)/(a_(n-2)))=(a_(n-1))/(a_(2))+(a_(2))/(a_(n-1)) .

If a_(1),a_(2),a_(3)"....." are in GP with first term a and common ratio r, then (a_(1)a_(2))/(a_(1)^(2)-a_(2)^(2))+(a_(2)a_(3))/(a_(2)^(2)-a_(3)^(2))+(a_(3)a_(4))/(a_(3)^(2)-a_(4)^(2))+"....."+(a_(n-1)a_(n))/(a_(n-1)^(2)-a_(n)^(2)) is equal to

If a_(a), a _(2), a _(3),…., a_(n) are in H.P. and f (k)=sum _(r =1) ^(n) a_(r)-a_(k) then (a_(1))/(f(1)), (a_(2))/(f (2)), (a_(3))/(f (n)) are in :

If a_(a), a _(2), a _(3),…., a_(n) are in H.P. and f (k)=sum _(r =1) ^(n) a_(r)-a_(k) then (a_(1))/(f(1)), (a_(2))/(f (2)), (a_(3))/(f (n)) are in :

RESONANCE ENGLISH-TEST PAPERS-MATHEMATICS
  1. cot^(-1)(-1/2)+cot^(-1)(-1/3) is equal to

    Text Solution

    |

  2. If the focus of a parabola is (3,3) and its directrix is 3x-4y=2 then ...

    Text Solution

    |

  3. If a(1),a(2),a(3),…a(n+1) are in arithmetic progression, then sum(k=0)...

    Text Solution

    |

  4. A line makes an angle theta with each of the x-and z-axes. If the an...

    Text Solution

    |

  5. The range of the function f(x)=2+x-[x-3] is, (where [.] denotes greate...

    Text Solution

    |

  6. about to only mathematics

    Text Solution

    |

  7. Statement 1: If e^(xy)+ln(xy)+cos(xy)+5=0, then (dy)/(dx)=-y/x. Stat...

    Text Solution

    |

  8. A spherical balloon is expanding. If the radius in increasing at the r...

    Text Solution

    |

  9. Let D(r) = |(a,2^(r),2^(16) -1),(b,3(4^(r)),2(4^(16) -1)),(c,7(8^(r)),...

    Text Solution

    |

  10. Let veca and vecb be unit vectors such that |veca+vecb|=sqrt3. Then fi...

    Text Solution

    |

  11. underset(xtooo)lim(e^(1//x^(2))-1)/(2tan^(-1)(x^(2))-pi) is equal to

    Text Solution

    |

  12. The equation of the tangent to the curve y=sqrt(9-2x^(2)) at the point...

    Text Solution

    |

  13. Let P and Q be two statements, then ~(~P^^Q)^^(PvvQ) is logically equi...

    Text Solution

    |

  14. If 4\ sin^(-1)x+cos^(-1)x=pi , then what is the value of x ?

    Text Solution

    |

  15. If sum(i=1)^(15)x(i)=45,A=sum(i=1)^(15)(x(i)-2)^(2) B=sum(i=1)^(15)(...

    Text Solution

    |

  16. If f(x)=sin|x|-e^(|x|) then at x=0,f(x) is

    Text Solution

    |

  17. If y=f((2x-1)/(x^2+1))andf^'(x)=sinx^2, then find (dy)/(dx)

    Text Solution

    |

  18. The no. of solutions of the equation a^(f(x)) + g(x)=0 where a>0 , and...

    Text Solution

    |

  19. Statement 1: If AnnB=phi then n((AxxB)nn(BxxA))=0 Statement 2: A-(Bu...

    Text Solution

    |

  20. Angle between diagonals of a prallelogramm whose sides are represented...

    Text Solution

    |