Consider the recurrence relation a_n = -a_{n-1} + 12a_{n-2} with first two terms a_0 = 3 and a_1 = 10. a. Write out the first 5 terms of the sequence defined by this recurrence relation. a_2 = 26, a_3 = 94, a_4 = 218, ... b. Solve the recurrence relation. That is, find a closed formula for a_n. a_n =
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Consider the recurrence relation $a_n = a_{n-1} + 12a_{n-2}$ with first two terms $a_0 = 3$ and $a_1 = a$. Write out the first 5 terms of the sequence defined by this recurrence relation. b. Solve the recurrence relation. That is, find a closed formula for $a_n$.
Adi S.
Consider the following recurrence relation and initial conditions. bk = 9bk-1 - 18bk-2, for every integer k ā„ 2 b0 = 2, b1 = 5 (a) Suppose a sequence of the form 1, t, t^2, t^3, ..., t^n ..., where t ā 0, satisfies the given recurrence relation (but not necessarily the initial conditions). What is the characteristic equation of the recurrence relation? What are the possible values of t? (Enter your answer as a comma-separated list.) (b) Suppose a sequence b0, b1, b2, ... satisfies the given initial conditions as well as the recurrence relation. Fill in the blanks below to derive an explicit formula for b0, b1, b2, ... in terms of n. It follows from part (a) and the single roots theorem that for some constants C and D, the terms of b0, b1, b2, ... satisfy the equation bn = for every integer n ā„ 0. Solve for C and D by setting up a system of two equations in two unknowns using the facts that b0 = 2 and b1 = 5. The result is that bn = for every integer n ā„ 0.
Identify the steps used to find the solution for the recurrence relation an = 2an-1 - 3 with the initial condition a0 = -1 using an iterative approach. (Check all that apply.) an = -3 + 2(-3) + 4an-2 an = -3 + 2(-3) + 4(-3) + 8an-3 an = -3 + 2(-3) + 4(-3) + 8(-3) + 16an-4 Continuing the same manner, an = -3(1 + 2 + 4 + ... + 2^{n-1}) + 2^nan-n = -3(2^n - 1) + 2^n(-1) = -2^{n+2} + 3 an = -3 - 3 + 2an-2 an = -6 + 2(-3) + 4an-3 an = -12 + 8an-4 Continuing the same manner, an = -3(1 + 2 + 4 + ... + 2^{n-1}) + an-n = -3(2^n - 1) + (-1) = -3 Ā· 2^n + 2
Madhur L.
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