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Question: If [x] denotes the greatest integer less than or equal to x and F = R – [R] where \(R = \left( 5\sqr...

If [x] denotes the greatest integer less than or equal to x and F = R – [R] where R=(55+11)2n+1R = \left( 5\sqrt{5} + 11 \right)^{2n + 1}then RF is equal to –

A

42n+1

B

42n

C

42n–1

D

None of these

Answer

42n+1

Explanation

Solution

Let G = (5511)2n+1(5\sqrt{5} - 11)^{2n + 1}, then 0 < G < 1,

Now, R – G = (55+11)2n+1(5\sqrt{5} + 11)^{2n + 1}(5511)2n+1(5\sqrt{5} - 11)^{2n + 1}

= 2 [2n+1C1(55)2n(11)+.....]\left\lbrack 2n + 1C_{1}\left( 5\sqrt{5} \right)^{2n}(11) + ..... \right\rbrack

= an even integer

Ž F – G = 0

Ž F = G

RF = RG = (55+11)2n+1(5511)2n+1\left( 5\sqrt{5} + 11 \right)^{2n + 1}\left( 5\sqrt{5} - 11 \right)^{2n + 1}

= 42n+1