Home
Class 12
MATHS
Let a1, a2, ..., an. be sequence of real...

Let `a_1, a_2, ..., a_n`. be sequence of real numbers with `a_(n+1) = a_n+ sqrt(1+a_n^2) and a_0 = 0`. Prove that `lim_(n->oo) (a_n/2^(n-1))=2/pi`.

Promotional Banner

Topper's Solved these Questions

  • LIMIT

    MOTION|Exercise EXERCISE -1|48 Videos

  • LIMIT

    MOTION|Exercise EXERCISE -2( LEVEL-I)|16 Videos

  • INVERSE TRIGONOMETRIC FUNCTIONS

    MOTION|Exercise Exercise -4 Level -II|7 Videos

  • MATRICES

    MOTION|Exercise Exercise - 4 (Level-II)|28 Videos

Similar Questions

Explore conceptually related problems

Let a_(1),a_(2),...,a_(n) .be sequence of real numbers with a_(n+1)=a_(n)+sqrt(1+a_(n)^(2)) and a_(0)=0 Prove that lim_(n rarr oo)((a_(n))/(2^(n-1)))=(2)/(pi)

Let a_(1),a_(2),a_(n) be sequence of real numbers with a_(n+1)=a_(n)+sqrt(1+a_(n)^(2)) and a_(0)=1 . Prove that lim_(xtooo)((a_(n))/(2^(n)))=4/(pi)

If a_1, a_2,......,a_n are n distinct odd natural numbers not divisible by any prime greater than 5, then prove that (1)/(a_1)+(1)/(a_2)+…..+(1)/(a_n) lt 2 .

A sequence a_1,a_2,a_3,.a_n of real numbers is such that a_1=0, |a_2|=|a_1+1|,|a_3|=|a_2+1|,gt,|an|=|a_(n-1)+1| . Prove that the arithmetic mean (a_1+a_2+……..+a_n)/n of these numbers cannot be les then - 1/2.

"If "a_1,a_2,a_3,.....,a_n" are in AP, prove that "a_(1)+a_(n)=a_(r)+a_(n-r+1)""

If the sequence {a_n}, satisfies the recurrence, a_(n+1) = 3a_n - 2a_(n-1), n ge2, a_0 = 2, a_1 = 3, then a_2007 is

Let a_1,a_2,a_3 ......, a_n are in A.P. such that a_n = 100 , a_40-a_39=3/5 then 15^(th) term of A.P. from end is

The n^(th) term, an of a sequence of numbers is given by the formula a_n = a_(n – 1) + 2n for n ge 2 and a_1 = 1 . Find an equation expressing an as a polynomial in n. Also find the sum to n terms of the sequence.

MOTION-LIMIT-EXERCISE-4
  1. Let a1, a2, ..., an. be sequence of real numbers with a(n+1) = an+ sqr...

    Text Solution

    |

  2. Let alpha and beta be the distinct roots of ax^(2) + bx + c = 0 then u...

    Text Solution

    |

  3. underset( x rarr 2 ) ( "lim") (( sqrt( 1- cos { 2 ( x - 2)}))/( x - 2...

    Text Solution

    |

  4. underset( x rarr 0 ) ( "lim")( (1- cos 2x) ( 3 + cos x ))/( x tan 4x) ...

    Text Solution

    |

  5. (lim)(xvec0)(sin(picos^2x))/(x^2) is equal to (1) pi/2 (2) 1 (3) ...

    Text Solution

    |

  6. Let p = underset( x to 0^(+)) lim(1 + tan^(2) sqrt(x))^(1//2x) , th...

    Text Solution

    |

  7. lim(x->pi/2)(cotx-cosx)/((pi-2x)^3)

    Text Solution

    |

  8. For each t in R let [t] be the greatest integer less than or equal to ...

    Text Solution

    |

  9. For x gt 0 , underset( x rarr 0) ( "Lim") [ ( sin x)^(1//x) + ((1)/( x...

    Text Solution

    |

  10. Let L = lim(x rarr 0) ( a - sqrt( a^2 - x^2) - x^2/4)/x^4, a>0.If L is...

    Text Solution

    |

  11. If underset( x to 0) lim [1+x log (1+b^(2))]^(1/x) = 2b sin^(2) the...

    Text Solution

    |

  12. If underset( x rarr oo) ( "lim") (( x^(2) + x+ 1)/(x+1) -ax -b) = 4, t...

    Text Solution

    |

  13. Let alpha(a) and beta(a) be the roots of the equation {(a+1)^(1//3)-1}...

    Text Solution

    |

  14. The largest value of non-negative integer a for which underset( x rarr...

    Text Solution

    |

  15. Let m and n be the two positive integers greater than 1 . If underset(...

    Text Solution

    |

  16. Let alpha, beta in R such that lim(x ->0) (x^2sin(betax))/(alphax-sin...

    Text Solution

    |

  17. Let f(x) = ((1 - x(1+ |1-x | )) /(|1-x|)) cos(1/(1-x)) for x!=1

    Text Solution

    |

  18. Let f: R rarr R be a differentiable function with f(0)=0. If y=f(x) ...

    Text Solution

    |