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If n is a positive integer, then: (sqr...

If n is a positive integer, then:
`(sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n)` is:

A

an irrational number

B

an odd positive integer

C

an even positive integer

D

a rational number other than positive integers

Text Solution

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The correct Answer is:
A
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Prove by induction that if n is a positive integer not divisible by 3 , then 3^(2n)+3^(n)+1 is divisible by 13 .

If n is a positive integer, which of the following two will always be integers: (I) (sqrt(2)+1)^(2n)+(sqrt(2)-1)^(2n) (II) (sqrt(2)+1)^(2n)-(sqrt(2)-1)^(2n) (III) (sqrt(2)+1)^(2n+1)+(sqrt(2)-1)^(2n+1) (IV) (sqrt(2)+1)^(2n+1)-(sqrt(2)-1)^(2n+1)

Knowledge Check

  • If n is a positive integer, then (1+i)^(n)+(1-i)^(n)=

    A
    `(sqrt(2))^(n-2) . cos ((n pi)/(4))`
    B
    `(sqrt(2))^(n-2) . sin ((n pi)/(4))`
    C
    `(sqrt(2))^(n+2) . cos ((n pi)/(4))`
    D
    `(sqrt(2))^(n+2) . sin ((n pi)/(4))`

  • The least positive integer n for which (sqrt(3)+i)^(n)=(sqrt(3)-i)^(n) is

    A
    3
    B
    4
    C
    6
    D
    8

  • If n is a positive integer, then n^(3)+2n is divisible

    A
    15
    B
    3
    C
    2
    D
    6

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