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Consider a sequence {an }with a1=2 and a...

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

A

162

B

64

C

32

D

2

Text Solution

Verified by Experts

The correct Answer is:
A, C, D
`(a_(n))/(a_(n-1))=(a_(n-1))/(a_(n-2))`
Here `a_(1),a_(2),a_(3)`…..are in G.P.
Let `a_(2)=x` then for `n=3`
`(a_(3))/(a_(2))=(a_(2))/(a_(1))impliesa_(2)^(2)=a_(1)a_(3)impliesa_(3)=(x^(2))/2`
`:.` G.P. is `2, x, (x^(2))/2, (x^(3))/4`,………..
Comon ratio `r=x/2`
Given `(x^(4))/8le162impliesx^(4)1296impliesxle6`
Also, `x` & `(x^(4))/8` are integers
so, if `x` is also even then only `(x^(4))/8` will be an integer
Hence, the possible values of `x` are 4 & 6, because `x!=2` as terms are distinct hence possible values of `a_(5)=(x^(4))/8` are `(4^(4))/8` & `(6^(4))/8`
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