Home
Class 12
MATHS
For a positive integer n let a(n)=1+1/2+...

For a positive integer `n` let `a(n)=1+1/2+1/3+1/4+.....+1/((2^n)-1)dot` Then

A

`a(100)lt100`

B

`a(100)gt100`

C

`a(200)gt100`

D

`a(200)lt100`

Text Solution

Verified by Experts

The correct Answer is:
A, C
`:.a(n)=1+(1)/(2)+(1)/(3)+(1)/(4)"......"+(1)/(2^(n)-1)`
`=1+((1)/(2)+(1)/(3))+((1)/(4)+(1)/(5)+(1)/(6)+(1)/(7))+((1)/(8)+"......"+(1)/(15))+"......"+(1)/(2^(n)-1)`
`=1+((1)/(2)+(1)/(2^(2)-1))+((1)/2^(2)+(1)/(5)+(1)/(6)+(1)/(2^(3)-1))+"......"+(1)/(2^(n)-1)`
`:. a(n)lt1+1+"....."+` n terms
`implies a(n)ltn`
`implies a(100)lt100`
Also, `a(n)=1+(1)/(2)+((1)/(3)+(1)/(4))+((1)/(5)+(1)/(6)+(1)/(7)+(1)/(8))+"......"+(1)/(2^(n)-1)`
`=1+(1)/(2)+((1)/(2^(1)+1)+(1)/(2^(2)))+((1)/(2^(2)+1)+(1)/(6)+(1)/(7)+(1)/(2^(3)))+"......"+((1)/(2^(n-1)+1)+"......"+(1)/(2^(n)))-(1)/(2^(n))`
`a(n)gt1+(1)/(2)+(2)/(4)+(4)/(8)+"......."+(2^(n-1))/(2^(n))-(1)/(2^(n))`
`a(n)gt(1-(1)/(2^(n)))+(n)/(2)" " implies a(n)gt(n)/(2)`
`:.a(200)gt100`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • SEQUENCES AND SERIES

    ARIHANT MATHS|Exercise Exercise (Passage Based Questions)|24 Videos

  • SEQUENCES AND SERIES

    ARIHANT MATHS|Exercise Exercise (Single Integer Answer Type Questions)|10 Videos

  • SEQUENCES AND SERIES

    ARIHANT MATHS|Exercise Exercise (Single Option Correct Type Questions)|30 Videos

  • PROPERTIES AND SOLUTION OF TRIANGLES

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|21 Videos

  • SETS, RELATIONS AND FUNCTIONS

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|12 Videos

Similar Questions

Explore conceptually related problems

For an positive integer n, prove that : i^(n) + i^(n+1) + i^(n+2) + i^(n+3) + i^(n+4) + i^(n + 5) + i^(n+6) + i^(n+7) = 0 .

Let n be a positive integer such that sin (pi/(2n))+cos(pi/(2n))=(sqrt(n))/2dot Then

Knowledge Check

  • For a positive integer ""^(n)C_(1) + ""^(n)C_(2) + ""^(n)C_(3) + …… + ""^(n)C_(n) is equal to:

    A
    `2^(n)`
    B
    `2^(n)-1`
    C
    `n^(2)`
    D
    `n^(2)-1`

    • Similar Questions

    Explore conceptually related problems

    The smallest positive integer n for which ((1+i)/(1-i))1

    The coefficient of x^n , where n is any positive integer, in the expansion of (1+2x+3x^2 +... to oo)^(1//2) is:

    Find the least positive integer n for which ((1+i)/(1-i))^n = 1

    Prove by induction that if n is a positive integer not divisible by 3 , then 3^(2n)+3^(n)+1 is divisible by 13 .

    By the Principle of Mathematical Induction, prove the following for all n in N : 1/1.3+1/3.5+1/5.7+......+ 1/((2n -1)(2n +1))= n/(2n +1) .

    If m and n are positive integers and f(x) = int_1^x(t-a )^(2n) (t-b)^(2m+1) dt , a!=b , then

    If, for a positive integer n , the quadratic equation, x(x+1)+(x-1)(x+2)+................+(x+ n-1)(x+n)=10 n has two consecutive integral solutions, then n is equal to : (1) 10 (2) 11 (3) 12 (4) 9

    Home
    Profile