Math Sequences and Recurrence Relations

28 Questions

To solve a second-order inhomogeneous difference equation, what is the first step?

Solve the associated homogeneous difference equation to get the complementary solution

What is the general form of the solution to a second-order inhomogeneous difference equation?

U_n = complementary solution + particular solution

What is the form of the particular solution when f(n) = 3^n?

b(a)^n

If the roots of the characteristic equation are distinct, what is the form of the complementary solution?

x(a^n)+y(b^n)

What is the next step after solving the characteristic equation?

Select the appropriate form of the particular solution

What is the purpose of solving the characteristic equation?

To find the complementary solution

What is the form of the particular solution when f(n) = an+b?

an+b

What is the final step in solving a second-order inhomogeneous difference equation?

Use terms in the sequence to find the missing coefficients

What is the purpose of equating the coefficients to get the particular solution?

To find the particular solution

What is the form of the solution to a second-order homogeneous difference equation?

U_n = x(a^n)+y(b^n)

What is the general formula for the nth term of a linear sequence?

T_n = a + (n - 1)d

In the recurrence relation T_n = T_(n+1) + 8, what is the value of T_2?

What is the difference between the terms in the sequence 9, 17, 25, 33?

What is the order of the difference equation u_n = 3u_(n-1) - 2?

First order

What is the general solution to the first order difference equation u_n = Au_(n-1) + B?

u_n = xAn + y

What is the condition for a sequence to be linear?

The first difference is constant

What is the value of x_3 in the sequence defined by the recurrence equation x_n = x_(n-1) - 1/5?

28/5

What is the formula to find the nth term of a linear sequence?

T_n = a + (n - 1)d

What is the general form of the solution to a second-order homogeneous difference equation with two distinct roots?

u_n=x(A^n)+y(B^n)

What is the difference between homogeneous and inhomogeneous difference equations?

Homogeneous equations have only terms involving previous values, while inhomogeneous equations have terms involving external values

How many equations are needed to solve for the coefficients in a second-order homogeneous difference equation with two distinct roots?

What is the general form of the solution to a second-order homogeneous difference equation with a double root?

u_n=x(A^n)+yn(A^n)

What is the purpose of forming a characteristic equation in solving a second-order homogeneous difference equation?

To find the roots of the equation

In a difference equation, what does a 15% increase in the population translate to mathematically?

An increase factor of 1.15

What is the form of the difference equation that models the deer population, given that 75 deer are killed per year?

u_n=1.15(u_(n-1))-75

What is the first step in solving a second-order homogeneous difference equation?

Form the characteristic equation

What is the purpose of substituting the given values into the solution of a second-order homogeneous difference equation?

To find the coefficients of the solution

What is the condition for a difference equation to be homogeneous?

All terms involve previous values

Study Notes

Recurrence Relations and Sequences

A recurrence relation is a formula that defines each term of a sequence using previous terms.
The recurrence relation T_n = T_(n+1) + 8 means that each term is 8 more than the previous term.

Linear Sequences

A linear sequence has a constant difference between consecutive terms.
The formula for a linear sequence is T_n = a + (n-1)d, where a is the first term and d is the common difference.
To find the nth term, substitute n into the formula.

First-Order Difference Equations

A first-order difference equation is a recurrence relation that defines each term using the previous term.
The general form of a first-order difference equation is u_n = A*u_(n-1) + B.
To solve a first-order difference equation, substitute u_n into the formula and solve for x and y.

Difference Equations

A difference equation is a recurrence relation that defines each term using previous terms.
Difference equations can be used to model real-life situations, such as population growth or financial modeling.

Second-Order Difference Equations

A second-order difference equation is a recurrence relation that defines each term using the previous two terms.
The general form of a second-order difference equation is u_n = A*u_(n-1) + B*u_(n-2) + C.
To solve a second-order difference equation, form the characteristic equation, find the roots, and select the appropriate form of the solution.

Homogeneous and Inhomogeneous Difference Equations

A homogeneous difference equation has only terms with u_n and its previous terms.
An inhomogeneous difference equation has terms with u_n and its previous terms, as well as additional terms.

Steps to Solve Second-Order Homogeneous Difference Equations

Form the characteristic equation.
Find the roots of the characteristic equation.
Select the appropriate form of the solution based on the roots.
Substitute the terms into the solution to find the missing coefficients.

Steps to Solve Second-Order Inhomogeneous Difference Equations

Solve the associated homogeneous difference equation to get the complementary solution.
Select the appropriate form of the particular solution.
Substitute the particular solution into the original equation for all u_n terms.
Equate the coefficients to get the particular solution.
Put the complementary and particular solutions together.
Use terms in the sequence to find the missing coefficients.

Choosing the Correct Particular Solution

Choose the correct form of the particular solution based on the form of the inhomogeneous term f(n).
Use a table to determine the correct form of the particular solution.

Solving Inhomogeneous Second-Order Difference Equations

Solve the associated homogeneous difference equation to get the complementary solution.
Select the appropriate form of the particular solution.
Substitute the particular solution into the original equation for all u_n terms.
Equate the coefficients to get the particular solution.
Put the complementary and particular solutions together.
Use terms in the sequence to find the missing coefficients.

Learn about recurrence relations and linear sequences, including formulas and definitions. Understand how to find the nth term and common difference.

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