Home
Class 12
MATHS
Let S=Sigma(n=1)^(999) (1)/((sqrt(n)+sqr...

Let `S=Sigma_(n=1)^(999) (1)/((sqrt(n)+sqrt(n+1))(4sqrt(n)+4sqrtn+1))` , then S equals ___________.

Text Solution

Verified by Experts

The correct Answer is:
9
Given `S=sum_(n=1)^(9999)1/((sqrtn+sqrn(n+1)(root4n+root(4)(n+1)))`
`=sum_(n=1)^(9999)1/((sqrtn+sqrt(n+1))(root4n+root(4)(n+1)))((root4n-root(4)(n+1))/(root4n-root(4)(n+1)))`
`=sum_(n=1)^(9999)((n+1)^(1//4)-n^(1//4))`
`=((2^(1/4)-1)+(3^(1/4)-2^(1/4))+(4^(1/4)-3^(1/4))+....+((9999+1)^(1/4)-(9999)^(1/4)))`
`=(10^(4))^(1/4)-1`
=9
Promotional Banner

Topper's Solved these Questions

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ARCHIVES ( JEE MAIN )(SINGLE CORRECT ANSWER TYPE )|16 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ARCHIVES ( JEE ADVANCED )(SINGLE CORRECT ANSWER TYPE )|3 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise EXERCIESE ( MATRIX MATCH TYPE )|3 Videos

  • PROBABILITY II

    CENGAGE PUBLICATION|Exercise MULTIPLE CORRECT ANSWER TYPE|6 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Archives (Numerical Value Type)|3 Videos

Similar Questions

Explore conceptually related problems

lim_(nto oo)[(sqrt(n+1)+sqrt(n+2)+...+sqrt(2)n)/(sqrt(n^(3)))]

lim_(nrarroo) [(1)/(sqrtn)+(1)/(sqrt(2n))+(1)/(sqrt(3n))+...+(1)/(n)]

The sum of the series (1)/(sqrt1+sqrt2)+ (1)/(sqrt2+ sqrt3)+ ...+ (1)/(sqrt(n)+sqrt(n+1)) is equal to-

lim_(nto oo)(sqrt1+sqrt2+......+sqrt(n-1))/(nsqrtn)=

If the value of lim_(n rarr oo)(n^(-3/2)) sum_(j=1)^(6n) sqrt(j) is equal to sqrt(N) , then the value of N/(12) is __________

underset(n to oo)lim(sqrt(1)+sqrt(2)+…+sqrt(n-1))/(nsqrt(n))=

If the value of ("lim")_(nvecoo)(n^(-3/2))dotsum_(j=1)^(6n)sqrt(j) is equal to sqrt(N) then the value of N/(8) is __________

The value of sum_(n=1)^(13)(i^(n)+i^(n+1)),i=sqrt(-1) is

The value of ("lim")_(nvecoo)sum_(r=1)^(4n)(sqrt(n))/(sqrt(r)(3sqrt(r)+sqrt(n))^2) is equal to 1/(35) (b) 1/4 (c) 1/(10) (d) 1/5

The value of the integral int_((1)/(n))^((an-1)/(n))(sqrt(x)dx)/(sqrt(a)-x+sqrt(x)) is -

CENGAGE PUBLICATION-PROGRESSION AND SERIES-EXERCIESE ( NUMERICAL VALUE TYPE )
  1. The terms a1, a2, a3 from an arithmetic sequence whose sum s 18. The t...

    Text Solution

    |

  2. Let the sum of first three terms of G.P. with real terms be 13/12 and ...

    Text Solution

    |

  3. The first term of an arithmetic progression is 1 and the sum of the fi...

    Text Solution

    |

  4. A person drops a ball from an 80 m tall building and each time the bal...

    Text Solution

    |

  5. Metals have conductivity in the order of ohm^(-1) cm^(-1)

    Text Solution

    |

  6. The number of positive integral ordered pairs of (a ,b) such that 6,a ...

    Text Solution

    |

  7. If the roots of 10 x^3-n x^2-54 x-27=0 are in harmonic oprogresi...

    Text Solution

    |

  8. Given a,b,c are in A.P.,b,c,d are in G.P and c,d,e are in H.P .If a=2 ...

    Text Solution

    |

  9. Let Sk be sum of an indinite G.P whose first term is 'K' and commmon r...

    Text Solution

    |

  10. The value of the sum Sigma(i=1)^(20) i(1/i+1/(i+1)+1/(i+2)+.....+1/(2)...

    Text Solution

    |

  11. The difference between the sum of the first k terms of the series 1^3+...

    Text Solution

    |

  12. The vlaue of the Sigma(n=0)^(oo) (2n+3)/(3^n) is equal to .

    Text Solution

    |

  13. The sum of the infinite Arithmetico -Geometric progression3,4,4,… is .

    Text Solution

    |

  14. Sigma(r=1)^(50)(r^2)/(r^2+(11-r)^2) is equal to .

    Text Solution

    |

  15. If Sigma(r=1)^(50) (2)/(r^2+(11-r^2)), then the value of n is

    Text Solution

    |

  16. Let lt an gt be an arithmetic sequence of 99 terms such that sum of it...

    Text Solution

    |

  17. Find the sum of series upto n terms ((2n+1)/(2n-1))+3((2n+1)/(2n-1))^2...

    Text Solution

    |

  18. Let S=Sigma(n=1)^(999) (1)/((sqrt(n)+sqrt(n+1))(4sqrt(n)+4sqrtn+1)) , ...

    Text Solution

    |

  19. Let S denote sum of the series 3/(2^3)+4/(2^4 .3)+5/(2^6 .3)+6/(2^7 .5...

    Text Solution

    |

  20. The sum (7)/(2^2xx5^2)+13/(5^2xx8^2)+19/(8^2xx11^2)+…10 terms is S, th...

    Text Solution

    |