Recurrence Relations (AQA A-Level Mathematics): Revision Notes

1. Identify the Recurrence Relation:

• Determine the type of relation (linear, homogeneous, or non-homogeneous).
• Identify the order of the relation (e.g., first-order, second-order).

2. Determine the Initial Conditions:

• Recurrence relations require initial terms to begin generating the sequence (e.g., $a_0$, $a_1$).

3. Check for Patterns:

• Substitution Method: Expand the recurrence for a few terms to observe any emerging patterns. This helps in forming a conjecture about the solution.
• Iteration: Express an​ as a function of earlier terms and try to write it explicitly.

4. Use the Characteristic Equation (if linear):

• For linear recurrence relations (e.g., $a_n$=$c_1a_{n−1}+c_2a_{n−2}$), set up the characteristic equation: $x_2−c_1x−c_2=0$
• Solve for the roots $r_1$ and $r_2$ to write the general solution based on the type of roots (real, distinct, or repeated).

5. Apply Initial Conditions:

• Substitute the initial conditions into the general solution to determine any constants.

6. For Non-Homogeneous Relations:

• Solve the homogeneous part first.
• Find a particular solution for the non-homogeneous part (e.g., using undetermined coefficients).
• Combine the two solutions for the final form.

Example:

The nth term of a sequence, $u_n$, is given by

Given that $u_3 = 11$,

a) Find the value of the constant $c$.

b) Find the value of $u_6$.

The terms of a sequence $u_1, u_2, u_3, \ldots$ are given by

where $k$ is a constant. Given that $u_1 = -4$,

a) Find expressions for $u_2$ and $u_3$ in terms of $k$.

b) Given also that $u_3 = 7u_2 + 3$, find the value of $k$.

c) Find the value of $u_4$.