Home
Class 12
MATHS
Define a sequence (a(n)) by a(1)=5,a(n)=...

Define a sequence `(a_(n))` by `a_(1)=5,a_(n)=a_(1)a_(2)…a_(n-1)+4` for `ngt1`. Then `lim_(nrarroo) (sqrt(a_(n)))/(a_(n-1))`

A

Equals `1//2`

B

equals 1

C

equals 2/5

D

does not exist

Text Solution

Verified by Experts

The correct Answer is:
C
`a_(1)=5`
`a_(n)=a_(1)a_(2)….a_(n-1)+4`
`a_(2)=a_(1)+4=9`
`a_(3)=a_(1).a_(2)+4=5xx9+4=49`
`a_(4)=a_(1)a_(2)a_(3)+4=2209`
`a_(5)=a_(1)a_(2)a_(3)a_(4)+4=4870849=(2207)^(2)`
`a_(5)=(a_(4)-2)^(2)`
`a_(4)=(49-2)^(2)=(a_(3)-2)^(2)`
`a_(3)=(9-2)^(2)=(a_(2)-2)^(2)`
`an=(a_(n-1)-2)^(2)`
`sqrt(a_(n))=a_(n-1)-2`
`(sqrt(a_(n)))/(a_(n-1))=(a_(n-1)-2)/(a_(n-1))=1-underset(nrarroo)"lim"(2)/(a_(n-1))`
`because underset(nrarroo)"lim"a_(n-1)=oo`
= 1
Promotional Banner

Topper's Solved these Questions

  • KVPY

    KVPY PREVIOUS YEAR|Exercise PART -I MATHEMATICS|20 Videos

  • KVPY

    KVPY PREVIOUS YEAR|Exercise PART-II MATHEMATICS|10 Videos

  • KVPY

    KVPY PREVIOUS YEAR|Exercise PART-2 MATHEMATICS|5 Videos

  • KVPY 2021

    KVPY PREVIOUS YEAR|Exercise PART II MATHEMATICS|4 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 sequence be defined by a_(1)=1,a_(2)=1 and a_(n)=a_(n-1)+a_(n-2) for all n>2, Find (a_(n+1))/(a_(n)) for n=1,2,3,4

If a_(1)=2 and a_(n)=2a_(n-1)+5 for ngt1 , the value of sum_(r=2)^(5)a_(r) is

The Sequence {a_(n)}_(n=1)^(+oo) is defined by a_(1)=0 and a_(n+1)=a_(n)+4n+3,n>=1 . Find the value of lim_(n rarr+oo)(sqrt(a_(n))+sqrt(a_(4n))+sqrt(a_(4^(2)n))+sqrt(a_(4^(3)n))+......+sqrt(a_(4^(10)n)))/(sqrt(a_(n))+sqrt(a_(2n))+sqrt(a_(2^(2)n))+sqrt(a_(2^(3)n))+.....+sqrt(a_(2^(10)n)))

find first 5 terms of the sequences, defined by a_(1) =-1 , a_(n) = ( a_(n-1))/n " for n ge 2

The fibonacct sequence is defined by 1=a_(1) = a_(2) and a_(n-1) + a_( n-1) , n gt 1 Find (a_(n+1))/(a_(n)) " for " n =1 ,2,34,5,

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)

The Fibonacci sequence is defined by a_(1)=1=a_(2),a_(n)=a_(n-2)f or n>2. Find (a_(n+1))/(a_(n))f or n=1,2,3,4,5