Let (5+2 √6)n=p+f where n ∈ N and p ∈ N and 0

Question

Let (5+2 √6)n=p+f where n ∈ N and p ∈ N and 0<f<1 then the value of, (f2-f+p f-p) is (A) a natural number (B) a negative integer (C) a prime number (D) an irrational number

Tardigrade Admin · Accepted Answer

sin / S f2-f+p f-p=-f(1-f)-p(1-f)=-[(p+f)(1-f)] where p + f =(5+2 √6) n and 1- f = f prime=(5-2 √6) n hence f 2- f + pf - p =-1.

Q. Let $(5 + 26)^{n} = p + f$ where $n \in N$ and $p \in N$ and $0 < f < 1$ then the value of, $(f^{2} - f + p f - p)$ is

a natural number

a negative integer

a prime number

an irrational number

$sin / S f^{2} - f + p f - p = - f (1 - f) - p (1 - f) = - [(p + f) (1 - f)]$
where $p + f = (5 + 26)^{n}$ and $1 - f = f^{'} = (5 - 26)^{n}$ hence $f^{2} - f + p f - p = - 1$ .

Q. Let (5+26​)n=p+f where n∈N and p∈N and 0<f<1 then the value of, (f2−f+pf−p) is