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Suppose-x-is-a-positive-real-number-such-that-x-x-and-x-are-in-a-geometric-progression-Find-the-least-positive-integer-n-such-that-x-n-gt-100-Here-x-denotes-the-integer-part-of-x-and-x-




Question Number 20652 by Tinkutara last updated on 30/Aug/17
Suppose x is a positive real number  such that {x}, [x] and x are in a  geometric progression. Find the least  positive integer n such that x^n  > 100.  (Here [x] denotes the integer part of x  and {x} = x − [x].)
Supposexisapositiverealnumbersuchthat{x},[x]andxareinageometricprogression.Findtheleastpositiveintegernsuchthatxn>100.(Here[x]denotestheintegerpartofxand{x}=x[x].)
Commented by ajfour last updated on 30/Aug/17
Is it n>(2/(log _(10) ((((√5)+1)/2))))  ⇒   (n)_(min) =10 .
Isitn>2log10(5+12)(n)min=10.
Commented by Tinkutara last updated on 30/Aug/17
Yes. But can it be solved in an exam  where calculators and log tables are  not allowed? I know the solution so no  need to post it.
Yes.Butcanitbesolvedinanexamwherecalculatorsandlogtablesarenotallowed?Iknowthesolutionsononeedtopostit.
Commented by ajfour last updated on 30/Aug/17
log ((((√5)+1)/2))=log (((2.236+1)/2))  =log (1.618)=log (16.18)−1            ≈4(0.3010)−1 ≈ 0.2  (2/(log ((((√5)+1)/2)))) ≈ 10 .
log(5+12)=log(2.236+12)=log(1.618)=log(16.18)14(0.3010)10.22log(5+12)10.
Commented by Tinkutara last updated on 31/Aug/17
Thanks!
Thanks!

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