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  • Let {x} and [x] denote the fractional part of x and the greatest integer  respectively of a real number x. If 

Let {x} and [x] denote the fractional part of x and the greatest integer \leq x respectively of a real number x. If \int_{0}^{n}\{x\} d x, \int_{0}^{n}[x] d x and 10(n^{2}-n), (n\epsilon N,n>1) are three consecutive terms of a G.P, then n is equal to_______
Option: 1 7
Option: 2 14
Option: 3 21
Option: 4 28
 

Answers (1)

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\\\int_{0}^{n}\{x\} d x=n \int_{0}^{1}\{x\} d x=n \int_{0}^{1} x d x=\frac{n}{2} \\ \int_{0}^{n}[x] d x=\int_{0}^{n}(x-\{x\}) d x=\frac{n^{2}}{2}-\frac{n}{2} \\ \Rightarrow\left(\frac{n^{2}-n}{2}\right)^{2}=\frac{n}{2} \cdot 10 \cdot n(n-1)(\text { where } n>1) \\ \Rightarrow \frac{n-1}{4}=5 \Rightarrow n=21

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himanshu.meshram

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