Assume that the sequence a = (a_n) satisfies the second-order recurrence relation a_{n+2} = 2a_n - a_{n+1} Find a closed-form formula for a_n if the first two terms are a_0 = 0 and a_1 = 1. Hint: follow the steps we took for solving the Fibonacci sequence. First, rewrite the recurrence relation in terms of operators E and I. Then factor this equation and find two basic solutions. The general solution will be some linear combination of these two basic solutions. a_n = 2 - (-2)^n a_n = 1/3 - 1/3(-2)^n a_n = (2 + ?5)^n / 4 - (2 - ?5)^n / 4 a_n = 2^n / 3 - (-1)^n / 3 a_n = 5^n / 9 - (-4)^n / 9 a_n = 5^n / 6 - (-1)^n / 6
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The given recurrence relation is $2a_n = a_{n-1} + a_{n-2}$. We can rewrite this in terms of the shift operator E and the difference operator [ as follows: $2(E^2 - E - 1)[a_n] = 0$ Step 2: Factor this equation. We can factor the polynomial $2(E^2 - E - 1)$ as Show more…
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