Sequences and Series for ESAT Advanced Mathematics

Definitions of Sequences and Progressions

In ESAT Mathematics, three specific terms are used to describe lists and sums of numbers: sequences, series, and progressions. A sequence is defined as an ordered list of numbers that often continues infinitely. A series is the result of summing the terms of a sequence. The term progression is used as a neutral, catch-all term that can refer to either sequences or series, and is often used in exam questions to maintain clarity.

Students are expected to write out terms of a sequence based on a given rule and to identify emerging patterns. These patterns are then used to make further mathematical deductions. The primary challenge is determining how many terms must be written before a pattern is reliably established.

Recurrence Relations and Pattern Spotting

Recurrence relations define each term based on previous terms. For example, consider the relation:

$x_0 = 1, x_1 = 2, x_{n+2} = |x_n - x_{n+1}|, n \geq 0$

Calculating the first four terms yields: $1, 2, 1, 1$. It might be tempting to assume the sequence repeats as $1, 2, 1, 1, 2, 1, 1...$ or remains constant as $1$. However, these conclusions are premature. Because the rule for $x_{n+2}$ depends on the two previous terms, we cannot conclude a pattern exists until we see a repeat in two consecutive terms.

Writing further terms: $1, 2, 1, 1, 0, 1, 1, 0, 1, 1...$

Here, the sequence stabilises into a repeating cycle of $(1, 1, 0)$. This demonstrates that the number of terms needed to justify a conclusion depends entirely on the structure of the recurrence relationship.

Summation and Sigma Notation

Sigma notation ($\Sigma$) is used to denote the sum of terms. When calculating a sum like $\sum_{n=0}^{100} x_n$, follow three key steps:

1. Identify the limits: In this case, the sum starts at $x_0$ and ends at $x_{100}$. This results in $101$ terms. This is the fence post issue: for any range from $m$ to $n$ inclusive, there are $n - m + 1$ terms.
2. Write out the first few terms: This helps determine the structure. For our example: $x_0 + x_1 + x_2 + x_3 + x_4 + x_5... = 1 + 2 + 1 + 1 + 0 + 1...$
3. Examine the end of the sum: Look at the final terms like $x_{98}, x_{99}, x_{100}$ to see how they fit into the observed blocks. In our example, the sequence consists of $x_0, x_1, x_2$ followed by repeating blocks of 3 terms $(1, 1, 0)$.

Arithmetic Progressions

An Arithmetic Progression (AP) is a sequence where the difference between consecutive terms is constant. You must be able to use and derive the following formulae:

• First term: $a$
• Common difference: $d = u_{n+1} - u_n$
• $n^{th}$ term: $u_n = a + (n - 1)d$
• Sum to $n$ terms: $S_n = \frac{n}{2}(2a + (n - 1)d) = \frac{n}{2}(u_1 + u_n)$

The sum can be understood as the number of terms multiplied by the average value of the terms: $S_n = n \times (\frac{u_1 + u_n}{2})$.

Linear combinations of arithmetic sequences are also arithmetic. If $u_n$ and $v_n$ are arithmetic sequences, then $S_n + T_n$ (the sum of their respective series) forms a new arithmetic series with first term $a + A$ and common difference $d + D$. This property holds for any linear combination $\alpha u_n + \beta v_n$.

Geometric Progressions

A Geometric Progression (GP) is a sequence where the ratio between consecutive terms is constant. The following formulae are essential:

• First term: $a$
• Common ratio: $r = \frac{u_{n+1}}{u_n}$
• $n^{th}$ term: $u_n = ar^{n-1}$
• Sum to $n$ terms: $S_n = \frac{a(1 - r^n)}{1 - r}$
• Sum to infinity: $S_\infty = \frac{a}{1 - r}$, which is valid only when $|r| < 1$

You can generate new GPs by transforming existing ones. For instance, if you have a GP with ratio $r$, replacing $r$ with $-r$ creates an alternating GP. Squaring every term in a GP with first term $a$ and ratio $r$ creates a new GP with first term $a^2$ and ratio $r^2$. Similarly, raising every term to the power of $k$ creates a GP with first term $a^k$ and ratio $r^k$.

Summing Parts of a Geometric Series

To find the sum of a specific portion of a GP, such as $ar^m + ar^{m+1} + \dots + ar^n$, use one of these three methods:

1. Difference of two sums: Calculate $S_{n+1} - S_m$. Note the index $n+1$ is required to include the $ar^n$ term.
2. Factorisation: Factorise out $r^m$ to get $r^m(a + ar + \dots + ar^{n-m})$. The term in the bracket is a standard GP starting at $a$.
3. New GP: Treat the sub-series as a completely new GP where the first term is $ar^m$, the common ratio is $r$, and the number of terms is $n - m + 1$.