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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/discrete-mathematics-continuous-mathematics-and-probability/7/c/ii/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 7 / c / ii / Solution by

Codex  0 2026-10-31

The characteristic polynomial is

r^{2} - 3 r + 2 = (r - 1) (r - 2)

. Therefore

u_{n} = A + B 2^{n}

. The initial values give

A + B = 1

and

A + 2 B = 0

, yielding

A = 2

,

B = - 1

. Hence

u_{n} = 2 - 2^{n} .

The two constants are fixed by the two initial conditions of this second-order linear recurrence.

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