Geometric Series and Sums for the ESAT

Understanding Progressions and Series

In the context of the ESAT, it is important to distinguish between three related terms. A sequence is an ordered list of numbers, which often continues indefinitely. A series is specifically the sum of the terms in a sequence. A progression is a more neutral catch-all term that examiners often use to make questions easier to understand by referring to the general pattern of terms.

When working with sequences, you must be able to write out terms given a rule, spot patterns, and make deductions. A crucial skill is determining how many terms are necessary to justify a conclusion about a pattern. For a recurrence relation like $x_{n+2} = |x_n - x_{n+1}|$, a term depends on the previous two, so you would need to see a repeat in two adjacent terms before a pattern is truly established.

The Geometric Progression

A Geometric Progression (GP) is a sequence where the ratio of adjacent terms is fixed. This constant is called the common ratio, denoted by $r$. If the first term is $a$, then the sequence is $a, ar, ar^2, ar^3, \dots, ar^{n-1}$.

The standard formulae you must know are:

1. The $n^{\text{th}}$ term: $u_n = ar^{n-1}$
2. The sum of the first $n$ terms: $S_n = \frac{a(1 - r^n)}{1 - r}$
3. The sum to infinity: $S_\infty = \frac{a}{1 - r}$, which is only valid when $|r| < 1$

You should also be comfortable with the relationship $\frac{u_{n+1}}{u_n} = r$, which implies $\frac{u_{n+1}}{u_n} = \frac{u_n}{u_{n-1}}$.

Sigma Notation and Indexing

Fluent use of $\Sigma$ (sigma) notation is expected. Note that the same sum can be represented in multiple ways by shifting the indices:

$S_n = \sum_{k=1}^{n} ar^{k-1} = \sum_{k=0}^{n-1} ar^k$

You must pay close attention to the limits of a sum. When summing from $n=0$ to $n=100$, there are actually $101$ terms. This is a common point of failure known as the fence post error. Always verify the number of terms by calculating $(\text{upper limit} - \text{lower limit}) + 1$.

Manipulating and Deriving New Progressions

You can generate new GPs by transforming an existing one. If you have a GP with first term $a$ and ratio $r$, consider these variations:

1. Alternating Ratios: Replacing $r$ with $-r$ gives $a - ar + ar^2 - ar^3 \dots$, which is a GP with common ratio $-r$. Its sum to infinity is $S_\infty = \frac{a}{1+r}$.
2. Linear Combinations: By taking $\frac{1}{2}(S_n + S_n(-r))$, you isolate the even-powered terms: $a + ar^2 + ar^4 + ar^6 \dots$. This is a new GP with first term $a$ and common ratio $r^2$.
3. Powers of Terms: If you square every term, you get $a^2 + a^2r^2 + a^2r^4 \dots$, which is a GP with first term $a^2$ and common ratio $r^2$. Similarly, raising every term to the power of $k$ creates a GP with first term $a^k$ and common ratio $r^k$.

Summing a Segment of a Geometric Series

You may be asked to find the sum of a specific section of a GP, such as $ar^m + ar^{m+1} + \dots + ar^n$ (where $n > m$). There are three reliable methods to calculate this:

Method 1: Difference of two sums Treat the segment as the difference between two sums starting from the first term. To get terms $m$ through $n$, take the sum of the first $n+1$ terms and subtract the sum of the first $m$ terms: $S_{n+1} - S_m$. Note the $n+1$ index is necessary because $S_{n+1}$ includes the $u_n$ term.

Method 2: Factorisation Factor out the common factor $ar^m$ to reveal a standard GP inside the brackets: $ar^m(1 + r + r^2 + \dots + r^{n-m})$.

Method 3: New Progression Treat the segment as a completely new GP. In this new GP, the first term is $ar^m$, the common ratio is $r$, and the number of terms is $n - m + 1$. Plugging these into the $S_n$ formula yields the same result as the other methods.