Home
Class 12
MATHS
The greatest interger by which 1+Sigma(...

The greatest interger by which `1+Sigma_(r=1)^(30) rxxr !` is divisible is

A

composite number

B

odd number

C

divisible by 3

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
D
`rxxr!=(r+1-1)xxr!`
`=(r+1)!-r!`
`=V(r )-V(r-1)`
`rArrsum_(r=1)^(30)r(r!)=V(31)-V(0)=(31)!-1`
`rArr1+sum_(r=1)^(30)r(r!)=31!`
which is divisble by 31, which are prime numbers.
Promotional Banner

Topper's Solved these Questions

  • PROGRESSION AND SERIES

    CENGAGE|Exercise EXERCIESE ( MULTIPLE CORRECT ANSWER TYPE )|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE|Exercise Exercise (Multiple & Comprehension)|65 Videos

  • PROGRESSION AND SERIES

    CENGAGE|Exercise Exercise 5.9|9 Videos

  • PROBABILITY II

    CENGAGE|Exercise NUMARICAL VALUE TYPE|2 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE|Exercise JEE Advanced Previous Year|11 Videos

Similar Questions

Explore conceptually related problems

The greatest integer by which 1+sum_(r=1)^(30)r xx r! is divisible is a.composite number b.odd number c.divisible by 3 d.none of these

Evaluate sum_(r=1)^(n)rxxr!

Find the sum of the series (sum_(r=1)^(n) rxxr !)

Sum of the series Sigma_(r=1)^(n) rlog((r+1)/(r)) is

Find the sum Sigma_(r=1)^(oo) (r)/(r^4+1/4)

The value of Sigma_(r=1)^(n) (a+r+ar)(-a)^r is equal to

Find the greatest number of four digits which is exactly divisible by each 8, 12, 18 and 30.

CENGAGE-PROGRESSION AND SERIES-Exercise (Single)
  1. Consider the sequence 1,2,2,4,4,4,8,8,8,8,8,8,8,8,... Then 1025th term...

    Text Solution

    |

  2. The value of Sigma(i=1)^(n) Sigma(j=1)^(i) underset(k=1)overset(j) =22...

    Text Solution

    |

  3. If 1^2+2^2+3^2++2003^2=(2003)(4007)(334) and (1)(2003)+(2)(2002)+(3)(2...

    Text Solution

    |

  4. If tn denotes the nth term of the series 2+3+6+11+18+….. Then t50 is

    Text Solution

    |

  5. The sum of series Sigma(r=0)^(r) (-1)^r(n+2r)^2 (where n is even) is

    Text Solution

    |

  6. If(1^2-t1)+(2^2-t2)+….+(n^2-tn)=(n(n^2-1))/(3) then tn is equal to

    Text Solution

    |

  7. If (1+3+5++p)+(1+3+5++q)=(1+3+5++r) where each set of parentheses cont...

    Text Solution

    |

  8. If Hn=1+12++1/ndot , then the value of Sn=1+3/2+5/3++(99)/(50) is H(50...

    Text Solution

    |

  9. The sum to 50 terms of the series 3/1^2+5/(1^2+2^2)+7/(1^+2^2+3^2)+...

    Text Solution

    |

  10. Let S=4/19+44/(19)^2+444/(19)^3+...oo then find the value of S

    Text Solution

    |

  11. If 1-1/3+1/5-1/7+1/9-1/(11)+=pi/4 , then value of 1/(1xx3)+1/(5xx7)+1/...

    Text Solution

    |

  12. If 1/(1^2)+1/(2^2)+1/(3^2)+ tooo=(pi^2)/6,t h e n1/(1^2)+1/(3^2)+1/(5^...

    Text Solution

    |

  13. lim(nrarroo) Sigma(r=1)^(n) (r)/(1xx3xx5xx7xx9xx...xx(2r+1)) is equal ...

    Text Solution

    |

  14. The greatest interger by which 1+Sigma(r=1)^(30) rxxr ! is divisible...

    Text Solution

    |

  15. If Sigma(r=1)^(n) r^4=I(n), " then "Sigma(r=1)^(n) (2r -1)^4 is equal ...

    Text Solution

    |

  16. Value of (1+1/3)(1+1/(3^2))(1+1/(3^4))(1+1/(3^8))oo is equal to 3 b. 6...

    Text Solution

    |

  17. If x1,x2 …,x(20) are in H.P and x1,2,x(20) are in G.P then Sigma(r=1)^...

    Text Solution

    |

  18. The value of Sigma(r=1)^(n) (a+r+ar)(-a)^r is equal to

    Text Solution

    |

  19. The sum of series x/(1-x^2)+(x^2)/(1-x^4)+(x^4)/(1-x^8)+ to infinite t...

    Text Solution

    |

  20. The sum of 20 terms of the series whose rth term s given by kT(n)=(-1)...

    Text Solution

    |