Sequences

Introduction to Sequences

Welcome to the world of Sequences! At its simplest, a sequence is just a list of numbers that follows a specific rule. Whether you are looking at the petals on a flower, the way a bank account grows with interest, or even the timing of a bouncing ball, you are looking at sequences. In this chapter, we will learn how to describe these patterns, find specific numbers in a long list without counting them all, and even add them up. Don't worry if it feels like a lot of formulas at first—we will break them down into simple steps!

1. The Basics: What is a Sequence?

A sequence is a set of numbers written in a particular order. Each number in the sequence is called a term. We usually use the notation \(a_k\) to represent the kth term (for example, \(a_1\) is the first term, \(a_2\) is the second, and so on).

Finite vs. Infinite:
• A finite sequence has a specific end point. (Example: 2, 4, 6, 8)
• An infinite sequence goes on forever. (Example: 2, 4, 6, 8, ...)

Generating a Sequence:
There are two main ways to "build" a sequence:
1. Position-to-term (Deductive): A formula where you plug in the position \(k\) to get the value. Example: \(a_k = 2 + 3k\). If you want the 5th term, you just calculate \(2 + 3(5) = 17\).
2. Term-to-term (Recurrence Relation): A rule that tells you how to get the next term from the current one. Example: \(a_{k+1} = a_k + 3\), starting with \(a_1 = 5\). This means "to get the next term, add 3 to the one you have."

Common Behaviors:
• Increasing: Every term is larger than the one before it.
• Decreasing: Every term is smaller than the one before it.
• Periodic: The terms repeat in a cycle (like a playlist on "repeat"). Example: 1, 0, -1, 1, 0, -1, ...
• Convergent: The terms get closer and closer to a specific value (the limit).
• Divergent: The terms do not settle toward a single value; they might go to infinity or bounce around.

Key Takeaway:

A sequence is just a list. You can find terms using a direct formula or by following a "next step" rule (recurrence).

2. Series and Sigma Notation

When we add the terms of a sequence together, we call it a series.

To save time writing long sums, mathematicians use the Greek letter Sigma (\(\Sigma\)). This is just a fancy instruction that means "Add them all up!"

Example: \(\sum_{r=1}^{n} r = 1 + 2 + 3 + ... + n\)

How to read it:
• The number at the bottom is where you start (usually \(r=1\)).
• The number at the top is where you stop (\(n\)).
• The expression in the middle is the rule for each term.

3. Arithmetic Progressions (AP)

An Arithmetic Progression (or arithmetic sequence) is one where you add or subtract the same amount every time. This amount is called the common difference (\(d\)).

Key Terms:
• \(a\) = First term
• \(d\) = Common difference
• \(n\) = The number of terms
• \(l\) = The last term

The Formulas you need:
1. The nth term: \(a_n = a + (n - 1)d\)
2. The Sum of n terms (\(S_n\)):
\(S_n = \frac{n}{2}(2a + (n - 1)d)\)
...or if you know the last term:
\(S_n = \frac{n}{2}(a + l)\)

Did you know?
The sum of the first \(n\) natural numbers (1 + 2 + 3 + ... + n) is a famous arithmetic series. The formula is \(S_n = \frac{n(n+1)}{2}\). Legend says a young mathematician named Gauss figured this out in primary school to annoy his teacher!

Quick Review - Arithmetic:

• If you are adding/subtracting, it's an AP.
• Mistake to avoid: When calculating \(d\), always do (Term 2 - Term 1). If the sequence is decreasing, \(d\) must be negative!

4. Geometric Progressions (GP)

A Geometric Progression is one where you multiply by the same amount every time. This multiplier is called the common ratio (\(r\)).

Key Terms:
• \(a\) = First term
• \(r\) = Common ratio

The Formulas you need:
1. The nth term: \(a_n = ar^{n-1}\)
2. The Sum of n terms (\(S_n\)):
\(S_n = \frac{a(1 - r^n)}{1 - r}\) (This is usually easier to use if \(|r| < 1\))
\(S_n = \frac{a(r^n - 1)}{r - 1}\) (This is usually easier to use if \(|r| > 1\))

Sum to Infinity (\(S_{\infty}\))

Don't worry if this seems tricky at first! Imagine you are standing 2 meters from a wall. You walk half the distance (1m), then half the remaining distance (0.5m), then half again (0.25m). You will get closer and closer to the wall, but theoretically, you’ll never pass it. The total distance you walk "tends toward" 2 meters.

A geometric series converges (has a sum to infinity) ONLY if the common ratio \(r\) is between -1 and 1 (written as \(|r| < 1\)).

The formula is: \(S_{\infty} = \frac{a}{1 - r}\)

Quick Review - Geometric:

• If you are multiplying, it's a GP.
• Common Ratio: Find \(r\) by dividing (Term 2 ÷ Term 1).
• Sum to infinity: Only exists if the numbers are getting smaller and heading toward zero (\(|r| < 1\)).

5. Modelling with Sequences

In the real world, we use sequences to predict things.

• Arithmetic models: Useful for things with constant growth, like saving a fixed amount of £10 every week.
• Geometric models: Useful for things that grow by a percentage, like population growth or compound interest. If a population grows by 5%, then \(r = 1.05\).

Step-by-Step for Modelling Problems:
1. Identify if it is Arithmetic (adding) or Geometric (multiplying).
2. Write down your "knowns": what is \(a\)? what is \(d\) or \(r\)?
3. Decide what the question is asking: Are you finding a specific term (\(a_n\)) or a total amount (\(S_n\))?
4. Plug the numbers into the correct formula and solve.

Common Pitfall:

Be careful with the value of \(n\). If you are looking at the value of an investment after 5 years, is that the 5th term or the 6th term? Usually, if the start of Year 1 is \(a\), then the end of Year 5 is the 5th term. Always check if the sequence starts at "Time 0" or "Time 1".

Summary Checklist

Check your understanding:
• Can I tell the difference between a sequence (list) and a series (sum)?
• Do I know how to find the nth term of an AP and a GP?
• Can I use Sigma notation correctly?
• Do I remember that a Sum to Infinity only exists if \(|r| < 1\)?
• Am I using the correct formula for the context of the problem?