8.2 MTH 288

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Solve these recurrence relations together with the initial conditions given. Arrange the steps to their corresponding step numbers to solve the recurrence relation an + 2 = -4an + 1+ 5an for n ≥ 0 together with the initial conditions a0 = 2 and a1 = 8.

1. The characteristic equation and its roots are r2 + 4r - 5 = 0 and r = -5, 1, respectively. 2. The general solution is an = α1(-5)n + α21n = α1(-5)n + α2. 3. Using initial conditions, 2 = α1 + α2 and 8 = -5α1 + α2 4. After solving, α1 = -1 and α2 = 3. Therefore, an = -(-5)n + 3.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Solve these recurrence relations together with the initial conditions given. Match the steps (in the right column) to their corresponding step numbers (in the left column) to solve the recurrence relation an = -6an − 1 - 9an − 2 for n ≥ 2 together with the initial conditions a0 = 3 and a1 = -3. 1. Step 1 2. Step 4 3. Step 2 4. Step 3

1. The characteristic equation and its roots are r2 + 6r + 9 = 0 and r = -3, -3, respectively. 2. After solving, α1 = 3 and α2 = -2. Therefore, an = 3 · (-3)n -2n(-3)n = (3 - 2n)(-3)n. 3. The general solution is an = α1(-3)n + α2n(-3)n. 4. Using initial conditions, 3 = α1 and -3 = -3α1 - 3α2

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Solve these recurrence relations together with the initial conditions given. Arrange the steps to solve the recurrence relation an = an − 2 for n ≥ 2 together with the initial conditions a0 = 5 and a1 = -1 in the correct order.

1. r2 − 1 = 0; r = -1, 1 2. an = α1(-1)n + α21n = α1(-1)n + α2 3. 5 = α1 + α2 -1 = -α1 + α2 4. α1 = 3 and α2 = 2Therefore, an = 3 · (-1)n + 2.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Solve these recurrence relations together with the initial conditions given. Arrange the steps to solve the recurrence relation an = 7an − 1 - 10an − 2 for n ≥ 2 together with the initial conditions a0 = 2 and a1 = 1 in the correct order.

1. r2 − 7r + 10 = 0 and r = 2, 5 2. an = α12n + α25n 3. 2 = α1 + α21 = 2α1 + 5α2 4. α1 = 3 and α2 = -1Therefore, an = 3 · 2n - 5n.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Solve these recurrence relations together with the initial conditions given. Arrange the steps to solve the recurrence relation an = an − 1 + 6an − 2 for n ≥ 2 together with the initial conditions a0 = 3 and a1 = 6 in the correct order.

1. r2 − r − 6 = 0 and r = −2, 3 2. an = α1(−2)n + α23n 3. 3 = α1 + α26 = −2α1 + 3α2 4. α1 = 3 / 5 and α2 = 12 / 5Therefore, an = (3 / 5)(−2)n + (12 / 5)3n.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the nonhomogeneous linear recurrence relation an = 2an − 1 + 2n. Click and drag the given steps to their corresponding step names to show that an = n2n is a solution of the given recurrence relation in the correct order.

If an = n2^n for all n, then an-1 = (n-1)2^(n-1) the right-hand side of the given recurrence relation can be written as 2(n-1)2^(n-1) + 2n substituting both equations into the recurrence, we obtain (n-1) 2^n + 2n = n2^n

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the nonhomogeneous linear recurrence relation an = 2an − 1 + 2n. Identify the solution of the given recurrence relation with a0 = 2.

an = (n + 2)2n

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Solve these recurrence relations together with the initial conditions given. Identify the solution of the recurrence relation an = 6an − 1 - 8an − 2 for n ≥ 2 together with the initial conditions a0 = 4 and a1 = 10.

an = 3 · 2n + 4n

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Solve these recurrence relations together with the initial conditions given. Identify the solution of the recurrence relation an = 2an − 1 - an − 2 for n ≥ 2 together with the initial conditions a0 = 4 and a1 = 1.

an = 4 - 3n

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the nonhomogeneous linear recurrence relation an = 2an − 1 + 2n. Identify the set of all solutions of the given recurrence relation using the theorem given below. If {a(p)n} is a particular solution of the nonhomogeneous linear recurrence relation with constant coefficients an = c1an − 1 + c2an − 2 +· · ·+ckan − k + F(n), then every solution of the form {a(p)n + a(h)n}, where {a(h)n} is a solution of the associated homogeneous recurrence relation an = c1an − 1 + c2an − 2 +· · ·+ckan − k.

an = α(2)n + n(2)n

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Identify the properties of the given recurrence relations. an = an - 2

linear and homogeneous with constant coefficients and degree 2

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Identify the properties of the given recurrence relations. an = 3an - 1 + 4an - 2 + 5an - 3

linear and homogeneous with constant coefficients and degree 3

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Identify the properties of the given recurrence relations. an = an - 1 + an - 4

linear and homogeneous with constant coefficients and degree 4

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Identify the properties of the given recurrence relations. an = 2nan - 1 + an - 2

linear and homogeneous with nonconstant coefficients

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Identify the properties of the given recurrence relations. an = an - 1 + 2

linear and nonhomogeneous with constant coefficients

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Identify the properties of the given recurrence relations. an = an - 1 + n

linear and nonhomogeneous with constant coefficients

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Identify the properties of the given recurrence relations. an = a2n - 1 + an - 2

nonlinear and homogeneous with constant coefficients