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Question: Integral part of (5\(\sqrt{5}\)+ 11)<sup>2n+1</sup> is...

Integral part of (55\sqrt{5}+ 11)2n+1 is

A

Even

B

Odd

C

Cannot say anything

D

Neither even nor odd

Answer

Even

Explanation

Solution

(55\sqrt{5}+ 11)2n+1 = I + f & let (55\sqrt{5}–11)2n+1= f¢

Also I + f –f¢ = (5 5\sqrt{5}+ 11)2n+1 – (55\sqrt{5}–11)2n+1 = 2 (integer)

= Even integer

Hence f –f¢ = 2k – I is an integer

But –1 < f – f¢ < 1, therefore f – f¢ = 0

Ž I = 2k = even integer