CAT Exam Model Paper 5 with solutions for free online practice

Question : 5 of 60

Marks: +1, -0

If ‘n’ is a natural number then the greatest integer less than or equal to

( 5 + √ { 19 } ) ^ { n }

even,
odd.
even when ‘n’ is even and odd when ‘n’ is odd.
even when ‘n’ is odd and odd when ‘n’ is even.

Validate

Solution:

Let’s assume that T denotes the integral part and 'F’ denotes the fraction part of

( 5 + √ { 19 } ) ^ { n }

⇒\I + \F = 5 ^ { n } + ^ { n } C _ { 1 } .5 ^ { n - 1 } .√{ 19 } + ^ { n } C _ { 2 } . 5 ^ { n - 2 } .19 + ^ { n } C _ { 3 } . 5 ^ { n - 3 } .19 .√ { 19 } +....

Now

( 5 + √ { 19 } ) ^ { n }

is a proper fraction as

( 5 + √ { 19 } ) <1

Let's assume that F'

( 5 + √ { 19 } ) ^ { n }

⇒ \F' = 5 ^ { n } - ^ { n } C _ { 1 } .5 ^ { n - 1 } .√{ 19 } + ^ { n } C _ { 2 } . 5 ^ { n - 2 } .19 - ^ { n } C _ { 3 } . 5 ^ { n - 3 } .19 .√ { 19 } +....

Hence,

1 + F + F ' = 2 [ 5 ^ { n } + ^ { n } C _ { 2 } . 5 ^ { n - 2 }. 19 + ^ { n } C _ { 4 } .5 ^ { n - 4 }.19 ^ { 2 } + . . . ]

Evidently, F + F' = 1 as the sum of two proper fractions is always less than 2 and greater than 0. Also the right hand side is an integer and hence F + F' should also be an integer.
Therefore I + 1 is an even number and I is an odd number.

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