Solve these recurrence relations together with the initial conditions given. a) $a_n = a_{n−1} + 6a_{n−2}$ for $n \geq 2$, $a_0 = 3, a_1 = 6$. b) $a_n = 7a_{n−1} − 10a_{n−2}$ for $n \geq 2$, $a_0 = 2, a_1 = 1.$c) $a_n = 6a_{n−1} − 8a_{n−2}$ for $n \geq 2, a_0 = 4, a_1 = 10$. d) $a_n = 2a_{n−1} − a_{n−2}$ for $n \geq 2, a_0 = 4, a_1 = 1$. e) $a_n = a_{n−2}$ for $n \geq 2, a_0 = 5, a_1 = −1$. f) $a_n = −6a_{n−1} − 9a_{n−2}$ for $n \geq 2, a_0 = 3, a_1 = −3$. g) $a_{n+2} = −4a_{n+1} + 5a_n$ for $n \geq 0, a_0 = 2, a_1 = 8$

Solve the recurrence relation

$$a_n = 6a_{n−1} − 12a_{n−2} + 8a_{n−3}$$

with a₀ = −5, a₁ = 4, and a₂ = 88.

Find the solution to each of these recurrence relations and initial conditions. Use an iterative approach. a) $a_n = 3a_{n−1}, a_0 = 2$. b) $a_n = a_{n−1} + 2, a_0 = 3$. c) $a_n = a_{n−1} + n, a_0 = 1$. d) $a_n = a_{n−1} + 2n + 3, a_0 = 4$. e) $a_n = 2a_{n−1} − 1, a_0 = 1. f) a_n = 3a_{n−1} + 1, a_0 = 1$. g) $a_n = na_{n−1}, a_0 = 5. h) a_n = 2na_{n−1}, a_0 = 1$.

Determine which of these are linear homogeneous recurrence relations with constant coefficients. Also, find the degree of those that are. a) $a_n = 3a_{n−2}.$b) $a_n = 3.$ c) $a_n = a^2_{n−1}.$ d) $a_n = a_{n−1} + 2a_{n−3}.$ e) $a_n = a_{n−1}/n.$f) $a_n = a_{n−1} + a_{n−2} + n + 3.$ g) $a_n = 4a_{n−2} + 5a_{n−4} + 9a_{n−7}$

Determine which of these are linear homogeneous recurrence relations with constant coefficients. Also, find the degree of those that are.

$$a) a_n = 3a_{n−1} + 4a_{n−2} + 5a_{n−3}. b) a_n = 2na_{n−1} + a_{n−2}. c) a_n = a_{n−1} + a_{n−4}. d) a_n = a_{n−1} + 2. e) a_n = a^2_{n−1} + a_{n−2}. f) a_n = a_{n−2}. g) a_n = a_{n−1} + n.$$

a) Find a recurrence relation for the number of bit strings of length n that contain three consecutive 0s. b) What are the initial conditions? c) How many bit strings of length seven contain three consecutive 0s?

Express the solution of the linear nonhomogenous recurrence relation

$$a_n = a_{n−1} + a_{n−2} + 1$$

for n ≥ 2 where a0 = 0 and a1 = 1 in terms of the Fibonacci numbers.

a) Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one stair or two stairs at a time.

b) What are the initial conditions?

c) In how many ways can this person climb a flight of eight stairs?

What is the general form of the particular solution guaranteed to exist of the linear nonhomogeneous recurrence relation $a_n=6a_{n-1}-12a_{n-2}+8a_{n-3}+F(n)$ if a) $F(n)=n^2$? b) $F(n)=2^n$? c) $F(n)=n2^n$? d) $F(n)=(-2)^n$? e) $F(n)=n^2 2^n$? f) $F(n)=n^3(-2)^n$? g) $F(n)=3$?

Find the solution to

$$a_n = 2a_{n−1} + a_{n−2} − 2a_{n−3}$$

for n = 3, 4, 5, . . . , with a₀ = 3, a₁ = 6, and a₂ = 0.

Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions. a) $a_n = −2a_{n−1}, a_0 = −1$. b) $a_n = a_{n−1} − a_{n−2}, a_0 = 2, a_1 = −1$. c) $a_n = 3a^2_{n−1}, a_0 = 1$. d) $a_n = na_{n−1} + a^2_{n−2}, a_0 = −1, a_1 = 0$. e) $a_n = a_{n−1} − a_{n−2} + a_{n−3}, a_0 = 1, a_1 = 1, a_2 = 2$

Give a recursive definition of the sequence {aₙ}, n = 1, 2, 3, . . . if

$$a) a_n = 4n − 2. b) a_n = 1 + (−1)^n. c) a_n = n(n + 1). d) a_n = n^2.$$

Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions.

a) $a_n = 2 (a_n−1)$ , $a_1 = 2$ ( Just find its general formula )

Find the solution to

$$a_n = 7a_{n−2} + 6a_{n−3}$$

with

$$a_0 = 9, a_1 = 10, and a_2 = 32.$$