Recurrence Relation Sequence Problem

The sequence is linear because when you subtract two terms of the sequence, you get a constant difference.

To get a constant difference when subtracting two terms of the sequence.

Take away the highest term from the lowest term.

<p>The solution is in the form of <code>u_n=xAn+y</code>.</p> Signup and view all the answers

<p>When you subtract two terms of the sequence and you find a constant difference.</p> Signup and view all the answers

<p>The formula is <code>T_n=a+(n-1)d</code>.</p> Signup and view all the answers

<p>Because when you subtract two terms of the sequence, you get a constant difference of -1/5.</p> Signup and view all the answers

<p>Substitute terms from the sequence into the formula <code>u_n=xAn+y</code> and solve for x and y.</p> Signup and view all the answers

<p>u_n = x(A^n) + y(B^n)</p> Signup and view all the answers

<p>u_n = 1.15u_(n-1)</p> Signup and view all the answers

<p>2x^2 - x - 6 = 0</p> Signup and view all the answers

<p>u_n = x(A^n) + yn(A^n)</p> Signup and view all the answers

<p>u_n = 1.15u_(n-1) - 75</p> Signup and view all the answers

<p>u_n = Au_(n-1) + B</p> Signup and view all the answers

<p>3x - 4 = 0</p> Signup and view all the answers

<p>To find the roots of the difference equation, which are used to form the solution.</p> Signup and view all the answers

<p>u_n = Au_(n-1) + Bu_(n-2) + f(n)</p> Signup and view all the answers

<p>A homogeneous difference equation has all terms containing previous terms, while an inhomogeneous difference equation has terms not containing previous terms.</p> Signup and view all the answers

<p>$x^2 - 2x - 3 = 0$</p> Signup and view all the answers

<p>$u_n = ax^n + bx^(n-1)$</p> Signup and view all the answers

<p>$2n + 8$</p> Signup and view all the answers

<p>$b(2^n)$</p> Signup and view all the answers

<p>$rac{1}{2}(3^n)$</p> Signup and view all the answers

<p>$u_n = 2(5^n) - 9(2^n) + 2n + 8$</p> Signup and view all the answers

<p>$x^2 - 3x + 2 = 0$</p> Signup and view all the answers

<p>$u_n = au_(n-1) + bu_(n-2) + f(n)$</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>$u_n = x(5^n) + y(2^n)$</p> Signup and view all the answers

<p>u_n=1.15(u_(n-1))</p> Signup and view all the answers

<p>u_n=1.15(u_(n-1))-75</p> Signup and view all the answers

<p>9x^2-12x+4=0</p> Signup and view all the answers

<p>T_n=(5) (3/2)^n-1(n) (3/2)^n</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>u_n=x(A^n)+y(B^n)</p> Signup and view all the answers

<p>u_n=x(A^n)+yn(A^n)</p> Signup and view all the answers

<p>To find the roots of the characteristic equation, which are used to form the general solution to the difference equation.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>u_n=A(u_(n-1))+B</p> Signup and view all the answers

<p>u_n = xAn + y</p> Signup and view all the answers

<p>When you subtract two terms of a sequence and find a constant difference, then the sequence is linear.</p> Signup and view all the answers

<p>To find the constant difference</p> Signup and view all the answers

<p>By substituting terms from the sequence into the general form of the solution and solving for x and y.</p> Signup and view all the answers

<p>T_n = a + (n-1)d</p> Signup and view all the answers

<p>Because the sequence has a constant difference of 8 between consecutive terms.</p> Signup and view all the answers

<p>By taking away the highest term from the lowest term.</p> Signup and view all the answers

<p>u_n = A(u_(n-1)) + B</p> Signup and view all the answers

<p>x(r_1^n) + y(r_2^n)</p> Signup and view all the answers

<p>b(3^n)</p> Signup and view all the answers

<ol> <li>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</li> </ol> Signup and view all the answers

<p>x^2 - 3x + 2 = 0</p> Signup and view all the answers

<p>u_n = A^n(u_0) + (A^(n-1) + A^(n-2) + ... + 1)B</p> Signup and view all the answers

<p>Use the initial conditions to form a system of linear equations, and then solve for the coefficients</p> Signup and view all the answers

<p>u_n = a(u_(n-1)) + b(u_(n-2))</p> Signup and view all the answers

<p>an + b</p> Signup and view all the answers

<ol> <li>Solve the characteristic equation, 2) find the general form of the solution, 3) use the initial conditions to find the coefficients</li> </ol> Signup and view all the answers

<p>To find the roots, which will help in finding the general form of the solution</p> Signup and view all the answers

<p>To define each term of a sequence using previous terms</p> Signup and view all the answers

<p>T_n = a + (n-1)d</p> Signup and view all the answers

<p>A recurrence relation that defines each term using the previous term</p> Signup and view all the answers

<p>u_n = A<em>u_(n-1) + B</em>u_(n-2) + C</p> Signup and view all the answers

<p>A homogeneous equation has terms with u_n and its previous terms, while an inhomogeneous equation has additional terms</p> Signup and view all the answers

<p>To solve a second-order homogeneous difference equation</p> Signup and view all the answers

<p>u_n = A<em>r^n + B</em>s^n</p> Signup and view all the answers

<p>To select the appropriate form of the solution</p> Signup and view all the answers

<p>Solve the associated homogeneous difference equation to get the complementary solution.</p> Signup and view all the answers

<p>To find the roots of the equation.</p> Signup and view all the answers

<p>u_n = A<em>u_(n-1) + B</em>u_(n-2)</p> Signup and view all the answers

<p>A homogeneous difference equation has only terms with u_n and its previous terms, while an inhomogeneous difference equation has terms with u_n and its previous terms, as well as additional terms.</p> Signup and view all the answers

<p>To select the appropriate form of the particular solution based on the form of the inhomogeneous term f(n).</p> Signup and view all the answers

<p>Put the complementary and particular solutions together.</p> Signup and view all the answers

<p>u_n = A*u_(n-1)</p> Signup and view all the answers

<p>To choose the correct form of the particular solution based on the form of the inhomogeneous term f(n).</p> Signup and view all the answers

<p>Bn</p> Signup and view all the answers

<p>T_n = a + (n-1)d</p> Signup and view all the answers

<p>u_n = A*u_(n-1) + B</p> Signup and view all the answers

<p>u_n = A<em>u_(n-1) + B</em>u_(n-2)</p> Signup and view all the answers

<p>To find the general form of the solution</p> Signup and view all the answers

<p>A homogeneous difference equation has only terms with u_n and its previous terms, while an inhomogeneous difference equation has additional terms</p> Signup and view all the answers

<p>u_n = A<em>r^n + B</em>s^n</p> Signup and view all the answers

<p>Solve the associated homogeneous difference equation and then find the particular solution</p> Signup and view all the answers

<p>u_n = A*2^n</p> Signup and view all the answers

<p>To match the form of the inhomogeneous term</p> Signup and view all the answers

<p>Put the complementary and particular solutions together</p> Signup and view all the answers

<p>A linear sequence has a constant difference between consecutive terms, whereas a first-order difference equation defines each term using the previous term.</p> Signup and view all the answers

<p>To find the roots, which are used to determine the form of the solution.</p> Signup and view all the answers

<p>u_n = A<em>u_(n-1) + B</em>u_(n-2) + C</p> Signup and view all the answers

<p>If it has only terms with u_n and its previous terms, it's homogeneous. If it has additional terms, it's inhomogeneous.</p> Signup and view all the answers

<p>They can accurately describe and predict population growth, financial modeling, and other phenomena.</p> Signup and view all the answers

<p>Find the roots, then select the appropriate form of the solution based on the roots.</p> Signup and view all the answers

<p>A recurrence relation defines each term using previous terms, while a difference equation is a specific type of recurrence relation.</p> Signup and view all the answers

<p>They help find the roots, which determine the form of the solution.</p> Signup and view all the answers

<p>A formula that defines each term of a sequence using previous terms.</p> Signup and view all the answers

<p>A homogeneous difference equation has only terms with u_n and its previous terms, while an inhomogeneous difference equation has additional terms.</p> Signup and view all the answers

<p>u_n = A<em>u_(n-1) + B</em>u_(n-2) + C.</p> Signup and view all the answers

<p>Form the characteristic equation, find the roots, and select the appropriate form of the solution.</p> Signup and view all the answers

<p>To find the roots, which are used to find the general form of the solution.</p> Signup and view all the answers

<p>A sequence with a constant difference between consecutive terms.</p> Signup and view all the answers

<p>To find the general formula for the nth term of a sequence.</p> Signup and view all the answers

<p>Based on the form of the inhomogeneous term f(n).</p> Signup and view all the answers

<p>u_n = A<em>u_(n-1) + B</em>u_(n-2)</p> Signup and view all the answers

<p>u_n = A*u_(n-1) + B.</p> Signup and view all the answers

<p>Form the characteristic equation, find the roots, and select the appropriate form of the solution.</p> Signup and view all the answers

<p>To find the general solution to the associated homogeneous difference equation.</p> Signup and view all the answers

<p>Solve the associated homogeneous difference equation, select the appropriate form of the particular solution, and find the general form of the solution.</p> Signup and view all the answers

<p>T_n = a + (n-1)d</p> Signup and view all the answers

<p>T_n = a + (n-1)d, where a is the first term and d is the common difference.</p> Signup and view all the answers

<p>Based on the form of the inhomogeneous term f(n).</p> Signup and view all the answers

<p>To find the general formula for the nth term of a sequence, including the effects of the inhomogeneous term.</p> Signup and view all the answers

<p>A polynomial equation that is used to find the roots of the difference equation.</p> Signup and view all the answers

<p>By using terms in the sequence and equating coefficients.</p> Signup and view all the answers

<p>A homogeneous difference equation has only terms with u_n and its previous terms, while an inhomogeneous difference equation has additional terms.</p> Signup and view all the answers