Sequences and Series

A sequence is a list of terms, while a series is the sum of the terms. A sequence can be defined in two ways:

A recurrence relation, e.g. (for the Fibonacci numbers)
A position-to-term formula, e.g. (for an arithmetic sequence). A sequence with a position-to-term formula can be denoted as , e.g. .

Sequences can be:

Periodic, meaning that there are repeated terms, e.g. . In this example, the period is 3.
Oscillating, meaning they are periodic with just two terms, e.g. .
Convergent, meaning the terms of the sequence get closer to a limiting value, e.g. the terms in converge to 0. A series can also be convergent, meaning the sum to infinity of the sequence has a finite value^[1]. For example, the sum of converges to 2.
Divergent, meaning the terms of the sequence are not convergent, e.g. . A series can also be divergent, meaning the sum to infinity of the sequence does not have a finite value.
Monotonically increasing, meaning each term is greater than the one before it (), e.g. . A monotonically decreasing sequence has each term less than the one before it ().

The sequence of Fibonacci numbers is defined by the recurrence relation , , , giving the first 5 terms as . The ratio of each term to the previous one converges to , the golden ratio.

The sequence of Lucas numbers^[2] is defined by the same recurrence relation as the Fibonacci numbers (each term is the sum of the two immediately before it), but and , giving the first 5 terms as .

Recurrence relations

A linear recurrence relation is where each term in a sequence is a linear function of previous terms, e.g. . A first-order recurrence relation is where each term only depends on the previous term and some function of , e.g. . For the AS content, only first-order recurrence relations will be considered.

A homogenous recurrence relation has the form , for some constant . In contrast, a non-homogenous recurrence relation has the form , where is some function of (in this course, a polynomial in or ).

A recurrence system is defined by the recurrence relation (e.g. ) and the initial conditions, e.g. . Our goal is typically to find a closed-form solution to the system: a position-to-term rule that only depends on . For the previous example, (you can verify this for ).

Solving recurrence relations

Solving recurrence relations takes a similar method to the method for second-order differential equations with constant coefficients. For a relation of form or , the solution will take the form:

Where is the complementary function and is the particular solution.

First-order recurrence relations

For a first-order recurrence relation, we start by finding the complementary function , which is related to the homogenous part of the recurrence relation:

Start with the reduced equation (only the homogeneous part).
Replace with (), giving the auxiliary equation.
Solve for , giving a complementary function of the form .
In the AS course, is always of the form . Note that if , this will always be a failure case.

Second-order recurrence relations

The method is similar to above but instead we obtain a quadratic in :

Start with the reduced equation (only the homogeneous part).
Replace with (), giving the auxiliary equation.
Solve for , giving two possible roots and .
The complementary function depends on the nature of the roots and :
For distinct real or complex roots and , .
For repeated real roots ,

Particular solution

We then find the particular solution , which is related to the non-homogenous part of the recurrence relation ():

Choose a form for based off :
- If ) is a number,
- If ) is linear,
- If ) is quadratic,
- If ) is an exponential (some constants and ),
A failure case occurs If the form of is the same as for . This requires multiplying by . It is possible for a new failure case to occur, requiring repeated multiplication by until resolved.
Substitute the appropriate form for into both sides of the relation and solve for , , and .

Then, apply the initial conditions to find remaining constants from the complementary function.

Footnote on convergence (outside of specification)
There are sequences where the terms converge to 0 that do not sum to a convergent series. For example, the harmonic sequence is convergent because the terms tend to 0. However, the harmonic series is not convergent.↩︎
Footnote on the Fibonacci and Lucas numbers
is more commonly used to denote the golden ratio than , but the specification uses .
Some definitions of the Fibonacci and Lucas numbers start from instead of (as an extra sidenote, Fibonacci himself started from 1 and 2).↩︎