Consider a sequence `{a_n }with a_1=2 and a_n=(a_(n-1)^ 2)/(a_(n-2))` for all `ngeq3,` terms of the sequence being distinct. Given that `a_1 and a_5` are positive integers and`a_5lt=162` then the possible value(s) of `a_5` can be (a) 162 (b) 64 (c) 32 (d) 2

Options

A. 162
B. 64
C. 32
D. 2

Answer

A, C

Explanation

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`