Binomial TheoremHard
Question
Let (5 + 2√6)n = p + f, where n ∈ N and p ∈ N and 0 < f < 1, then the value of f2 - f + pf - p is:
Options
A.a natural number
B.a negative integer
C.a prime number
D.an irrational number
Solution
(5 + 2√6)n = p + f
(5 - 2√6)n = f′ 0 < f + f′ < 2
p + f + f′ = 2 [integer]
so f + f′ = integer = 1
∵ n ∈ N
(f - 1)(p + f) = - f′ (p + f) = - (+1)n = - 1
(5 - 2√6)n = f′ 0 < f + f′ < 2
p + f + f′ = 2 [integer]
so f + f′ = integer = 1
∵ n ∈ N
(f - 1)(p + f) = - f′ (p + f) = - (+1)n = - 1
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