Sequences and Arithmetic Series for the ESAT

Understanding Sequences and Series

In the context of Advanced Mathematics, it is important to distinguish between three related terms: sequence, series, and progression. A sequence is an ordered list of numbers, which often continues indefinitely. A series is the sum of the terms in a sequence. The term progression is a more general, neutral term used in the ESAT to describe these patterns, helping to make question wording more accessible. For example, examiners might refer to the first term of a series when technically referring to the first term of the underlying sequence whose sum forms that series.

Generating Sequences and Spotting Patterns

Sequences can be defined by a formula for the $n^{\text{th}}$ term or by a recurrence relation of the form $x_{n+1} = f(x_n)$. To make deductions about a sequence, you must write out enough terms to justify your conclusions. The number of terms required depends on the structure of the recurrence relationship. If a term depends on the previous two terms, you must see a repeat in two adjacent terms before a pattern can be confirmed.

Consider the sequence defined by: $x_0 = 1, x_1 = 2$ and $x_{n+2} = |x_n - x_{n+1}|$ for $n \geq 0$.

The first few terms are: $x_0 = 1$ $x_1 = 2$ $x_2 = |1 - 2| = 1$ $x_3 = |2 - 1| = 1$ $x_4 = |1 - 1| = 0$ $x_5 = |1 - 0| = 1$ $x_6 = |0 - 1| = 1$ $x_7 = |1 - 1| = 0$

While the first four terms (1, 2, 1, 1) might suggest a simple repetition, continuing the sequence reveals a periodic pattern: 1, 2, 1, 1, 0, 1, 1, 0, 1, 1... After the initial terms, the sequence repeats the block (1, 0, 1) with a period of three.

Sigma Notation and the Fence Post Issue

When calculating the sum of a sequence, we use sigma ($\Sigma$) notation. It is vital to pay close attention to the limits. For a sum $\sum_{n=0}^{100} x_n$, there are 101 terms, not 100. This is known as the fence post error: to find the number of terms between $n = a$ and $n = b$ inclusive, the formula is $b - a + 1$.

To find the sum of the sequence above from $n=0$ to $n=100$, you should write out the terms to see how the patterns align with the indices: $\sum_{n=0}^{100} x_n = x_0 + x_1 + x_2 + (x_3 + x_4 + x_5) + (x_6 + x_7 + x_8) + \dots + (x_{99} + x_{100})$ By grouping the sequence into blocks of three, where each block starting at $x_{3k}$ (for $k \geq 1$) follows the pattern (1, 0, 1), the total sum becomes easy to compute.

Arithmetic Progressions (APs)

An arithmetic progression is a sequence where each term is found by adding a constant, known as the common difference ($d$), to the previous term. You must be able to recognise these sequences and use their standard formulae:

1. First term: $a$
2. Common difference: $d = u_{n+1} - u_n$
3. $n^{\text{th}}$ term: $u_n = a + (n - 1)d$
4. Sum to $n$ terms: $S_n = \frac{n}{2}(2a + (n - 1)d)$

The sum formula can also be expressed as $S_n = \frac{n}{2}(u_1 + u_n)$. This can be interpreted conceptually as: $S_n = \text{number of terms} \times \text{average value of terms}$

A special case is the sum of the first $n$ natural numbers ($1, 2, 3, \dots, n$). Here, $a = 1$ and $d = 1$, so the sum is: $S_n = \frac{n}{2}(1 + n)$

Combining and Modifying Arithmetic Sequences

You can create new arithmetic series by combining existing ones. If you have two arithmetic series $S_n$ (with first term $a$ and difference $d$) and $T_n$ (with first term $A$ and difference $D$), their sum $S_n + T_n$ forms a new arithmetic series. The first term of this new series is $a + A$ and the common difference is $d + D$. This property extends to any linear combination of arithmetic sequences, such as $\alpha u_n + \beta v_n$, which will also be an arithmetic sequence.