If `p=(8+3sqrt(7))^n a n df=p-[p],w h e r e[dot]` denotes the greatest integer function, then the value of `p(1-f)` is equal to `1` b. `2` c. `2^n` d. `2^(2n)`

Options

A. 1
B. 2
C. `2^(n)`
D. `2^(2n)`

Answer

A

Explanation

`p=(8+3sqrt(7))^(n)=.^(n)C_(0)8^(n)+.^(n)C_(1)8^(n-1)(3sqrt(7))+"...."`
Let `p_(1)=(8-3sqrt(7))^(n)=.^(n)C_(0)8^(n)-.^(7)C_(1)8^(n-1)(3sqrt(7))+"....."`
`p + p_(1) = 2(.^(n)C_(0)8^(n)+.^(n)C_(2)8^(n-2)(3sqrt(7))^(2)+".....")="even integer"`
`p_(1)` clearly belongs to `(0,1)`
`rArr [p] = f+p_(1) = "even integer"`
`rArr f + p_(1) = "integer"`
`f in (0,1),p_(1)in(0,1)`
`rArr f + p in (0,2)`
`rArr f+ p_(1) = 1`
`rArr p_(1)=1-f`
Now, `p(1-f)=pp_(1)=[(8+3sqrt(7))^(n)(8-3sqrt(7))]^(n)=1`