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  • If  denotes greatest integer <img alt="\leq x" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cleq%2

If [x] denotes greatest integer \leq x and g=(3+\sqrt{5})^n then  [g] must be :

Option: 1

(3+\sqrt{5})^n+(3-\sqrt{5})^n+1


Option: 2

(3+\sqrt{5})^n+(3-\sqrt{5})^n-1


Option: 3

(3+\sqrt{5})^n+(3-\sqrt{5})^n


Option: 4

(3+\sqrt{5})^n-(3-\sqrt{5})^n


Answers (1)

best_answer

g=(3+\sqrt{5})^n=[g]+f
Consider another fraction f^{\prime}=(3-\sqrt{5})^n
\begin{array}{ll} \Rightarrow \quad[g]+f+f^{\prime}=(3+\sqrt{5})^n(3-\sqrt{5})^n=\text { integer } \\ \text { and } \quad 0<f<1 \text { and } 0<f^{\prime}<1 \Rightarrow f+f^{\prime}=1 \\ \text { so } \quad[g]=(3+\sqrt{5})^n+(3-\sqrt{5})^n-1 \end{array}
 

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