 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability/2/e/ii/solution

F_{n} = A ϕ^{n} + B ψ^{n} .

A + B = 0

and

A ϕ + B ψ = 1

. Since

ϕ - ψ = 5

, we obtain

A = 1/ 5

and

B = - 1/ 5

F_{n} = \frac{ϕ ^{n} - ψ ^{n}}{5} = \frac{1}{5} [(\frac{1 + 5}{2})^{n} - (\frac{1 - 5}{2})^{n}] .

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