Question

Consider a sequence `{a_n}` with a_1=2 & `a_n =(a_(n-1)^2)/(a_(n-2))` for all ` n ge 3` terms of the sequence being distinct .Given that `a_2 " and " a_5` are positive integers and `a_5 le 162`, then the possible values (s) of `a_5` can be
A. 136
B. 64
C. 32
D. 2

Answer 1

Correct Answer - A::C
Given `a_(1)=2,(a_(n))/(a_(n-1))=(a_(n-1))/(a_(n-2))`
Hence,`a_(1),a_(2),a_(3),a_(4),a_(5),..` in G.P.
Let `a_(2)=x`. Then for n=3, we have
`(a_(3))/(a_(2))=(a_(2))/(a_(1))`
`rArra_(2)^(2)=a_(1)a_(3)`
`rArra_(3)=x^(2)/2`
i.e., `2,x,x^(2)/2,x^(3)/4,x^(4)/8`,... with common ratio `r=x/2`
Given `x^(4)/8` are integers.
So if x is even, then only `x^(4)/8` will be an integer.
Hence, possible value of x is 4 and 6. (`xne2` as terms are distinct)
Hence, possible value of `a_(5)=x^(4)/8` is `4^(4)/8,6^(4)/8`