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If (8 + 3sqrt(7))^(n) = P + F , where P...

If `(8 + 3sqrt(7))^(n) = P + F ` , where P is an integer and F is a proper fraction , then

A

P is a odd integer

B

P is an even integer

C

`F(P + F) = 1`

D

`(1 - F ) (P + F) = 1`

Text Solution

Verified by Experts

The correct Answer is:
a,d
We have , ` (8+ 3 sqrt(7))^(n) = (8 + sqrt(63))^(n)`
Now ,let ` P + F = (8 + sqrt(63))^(n)` …(i)
` 0 lt F lt 1` ….(ii)
and let ` F' = (8 - sqrt(63))^(n)` ….(iii)
` 0 lt F' lt 1 ` …(iv)
On adding Eqs (i) and (iii) , we grt
` P + F + F' = (8 + (sqrt(63))^(n) + (8 - sqrt(63))^(n)` ...(v)
` rArr P + 1 = 2p ` (even integer ) , ` AA p in ` N
` rArr P= 2p - 1 ` = odd integer
` therefore F' = 1 - F `
` therefore (1 - F ) (P - F ) = F' (P + F) = (8 - sqrt(63))^(n) (8 + sqrt(63))^(n)`
` = (64 - 63 )^(n) = 1^(n) = 1 `
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