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InterviewSolution
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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 beA. 136B. 64C. 32D. 2 |
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Answer» 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` |
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