56 Questions
Explain why the sequence generated by the recurrence relation T_n=T_(n+1)+8 is linear.
The sequence is linear because when you subtract two terms of the sequence, you get a constant difference.
What is the purpose of rearranging the recurrence relation x_n= x_(n-1)–1/5?
To get a constant difference when subtracting two terms of the sequence.
How do you find the order of a difference equation?
Take away the highest term from the lowest term.
What is the general form of the solution to a first-order difference equation in the form of u_n=A(u_(n-1))+B?
The solution is in the form of u_n=xAn+y.
How do you determine if a sequence is linear?
When you subtract two terms of the sequence and you find a constant difference.
What is the formula for the nth term of a linear sequence?
The formula is T_n=a+(n-1)d.
Explain why the recurrence relation x_n= x_(n-1)–1/5 generates a linear sequence.
Because when you subtract two terms of the sequence, you get a constant difference of -1/5.
How do you solve a first-order difference equation?
Substitute terms from the sequence into the formula u_n=xAn+y and solve for x and y.
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 equation that models the deer population if there is a 15% increase in the population each year?
u_n = 1.15u_(n-1)
What is the characteristic equation of the difference equation 2u_(n+1) - u_n - 6u_(n-1) = 0?
2x^2 - x - 6 = 0
What is the 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 difference equation that models the deer population if 75 deer are killed per year?
u_n = 1.15u_(n-1) - 75
What is the general form of a first-order difference equation?
u_n = Au_(n-1) + B
What is the characteristic equation of the difference equation 3u_(n+1) - 4u_n = 0?
3x - 4 = 0
What is the purpose of finding the characteristic equation of a difference equation?
To find the roots of the difference equation, which are used to form the solution.
What is the general form of a second-order inhomogeneous difference equation?
u_n = Au_(n-1) + Bu_(n-2) + f(n)
What is the difference between a homogeneous and inhomogeneous difference equation?
A homogeneous difference equation has all terms containing previous terms, while an inhomogeneous difference equation has terms not containing previous terms.
Given the difference equation $u_n = 2u_(n-1) + 3u_(n-2)$, what is the characteristic equation?
$x^2 - 2x - 3 = 0$
What is the general form of the complementary solution for a 2nd order homogeneous difference equation with a repeated root?
$u_n = ax^n + bx^(n-1)$
Given the difference equation $u_n - 7u_(n-1) + 10u_(n-2) = 6n + 8$, what is the particular solution?
$2n + 8$
What is the correct form of the particular solution for a 2nd order inhomogeneous difference equation with $f(n) = 2^n$?
$b(2^n)$
Given the difference equation $u_n = 3u_(n-1) - 2u_(n-2) + 3^n$, what is the particular solution?
$rac{1}{2}(3^n)$
What is the final solution to the difference equation $u_n - 7u_(n-1) + 10u_(n-2) = 6n + 8$, given that $u_0 = 1$ and $u_1 = 2$?
$u_n = 2(5^n) - 9(2^n) + 2n + 8$
What is the characteristic equation of the difference equation $u_n - 3u_(n-1) + 2u_(n-2) = 3^n$?
$x^2 - 3x + 2 = 0$
What is the general form of a 2nd order inhomogeneous difference equation?
$u_n = au_(n-1) + bu_(n-2) + f(n)$
How do you solve a 2nd order inhomogeneous difference equation?
Solve the associated homogeneous difference equation, select the appropriate form of the particular solution, substitute the particular solution into the original equation, equate coefficients to get the particular solution, put the complementary and particular solutions together, and use terms in the sequence to find the missing coefficients.
What is the complementary solution for the difference equation $u_n - 7u_(n-1) + 10u_(n-2) = 0$?
$u_n = x(5^n) + y(2^n)$
What is the difference equation that models the deer population if there is a 15% increase in the population each year?
u_n=1.15(u_(n-1))
How does the model change if 75 deer are killed per year?
u_n=1.15(u_(n-1))-75
What is the characteristic equation of the difference equation 9T_(n+1)-12T_n+4T_(n-1)=0?
9x^2-12x+4=0
What is the solution to the difference equation 9T_(n+1)-12T_n+4T_(n-1)=0 given T_0=5 and T_1=6?
T_n=(5) (3/2)^n-1(n) (3/2)^n
What is the difference between a homogeneous and inhomogeneous difference equation?
A homogeneous difference equation has every term being a multiple of a previous term, while an inhomogeneous difference equation has some terms not being a multiple of a previous term.
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 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 finding the characteristic equation of a difference equation?
To find the roots of the characteristic equation, which are used to form the general solution to the difference equation.
How do you solve a second-order homogeneous difference equation?
Form the characteristic equation, find the roots, select the appropriate form of the solution, and substitute in terms of the sequence to find the missing coefficients.
What is the general form of a first-order difference equation?
u_n=A(u_(n-1))+B
What is the general form of the solution to a first-order difference equation in the form of u_n=A(u_(n-1))+B?
u_n = xAn + y
How do you determine if a sequence is linear?
When you subtract two terms of a sequence and find a constant difference, then the sequence is linear.
What is the purpose of rearranging the recurrence relation x_n= x_(n-1)–1/5?
To find the constant difference
How do you solve a first-order difference equation?
By substituting terms from the sequence into the general form of the solution and solving for x and y.
What is the formula for the nth term of a linear sequence?
T_n = a + (n-1)d
Explain why the sequence generated by the recurrence relation T_n=T_(n+1)+8 is linear.
Because the sequence has a constant difference of 8 between consecutive terms.
How do you find the order of a difference equation?
By taking away the highest term from the lowest term.
What is the general form of a first-order difference equation?
u_n = A(u_(n-1)) + B
What is the general form of the solution to a second-order homogeneous difference equation with two distinct roots?
x(r_1^n) + y(r_2^n)
What is the correct form of the particular solution for a 2nd order inhomogeneous difference equation with f(n) = 3^n?
b(3^n)
How do you solve a second-order inhomogeneous difference equation?
- Solve the associated homogeneous difference equation, 2) select the appropriate form of the particular solution, 3) substitute the particular solution into the original equation, 4) equate the coefficients, 5) put the complementary and particular solution together, and 6) use terms in the sequence to find the missing coefficients
What is the characteristic equation of the difference equation u_n - 3u_(n-1) + 2u_(n-2) = 3^n?
x^2 - 3x + 2 = 0
What is the general form of the solution to a first-order difference equation in the form of u_n = A(u_(n-1)) + B?
u_n = A^n(u_0) + (A^(n-1) + A^(n-2) + ... + 1)B
How do you find the coefficients in the complementary solution?
Use the initial conditions to form a system of linear equations, and then solve for the coefficients
What is the general form of a second-order homogeneous difference equation?
u_n = a(u_(n-1)) + b(u_(n-2))
What is the correct form of the particular solution for a 2nd order inhomogeneous difference equation with f(n) = 2n + 8?
an + b
How do you solve a second-order homogeneous difference equation?
- Solve the characteristic equation, 2) find the general form of the solution, 3) use the initial conditions to find the coefficients
What is the purpose of finding the characteristic equation of a difference equation?
To find the roots, which will help in finding the general form of the solution
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) + 8means 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, whereais the first term anddis the common difference. - To find the
nth term, substituteninto 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_ninto the formula and solve forxandy.
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_nand its previous terms. - An inhomogeneous difference equation has terms with
u_nand 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_nterms. - 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_nterms. - 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.
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) + 8means 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, whereais the first term anddis the common difference. - To find the
nth term, substituteninto 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_ninto the formula and solve forxandy.
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_nand its previous terms. - An inhomogeneous difference equation has terms with
u_nand 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_nterms. - 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_nterms. - 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.
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) + 8means 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, whereais the first term anddis the common difference. - To find the
nth term, substituteninto 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_ninto the formula and solve forxandy.
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_nand its previous terms. - An inhomogeneous difference equation has terms with
u_nand 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_nterms. - 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_nterms. - 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.
Solve the recurrence relation Tn=T(n+1)+8 to find T100, given T1=9. Learn how to generate a sequence from a recurrence relation and apply it to find the 100th term.
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