Introduction to Sequences and Series

TL;DR

Sequences are ordered lists of numbers, while series are the sums of those numbers in a sequence. Understanding the patterns in sequences helps us predict future terms and calculate their sums. Arithmetic and geometric progressions are specific types of sequences with predictable rules for their terms.

1. The Mental Model

Think of sequences like steps on a ladder: each step is a term, and they follow a specific order. A series is like measuring the total height you've climbed after a certain number of steps.

2. The Core Material

What is a Sequence?

A sequence is just an ordered list of numbers. Each number in the list is called a term. We often use $u_n$ to represent the $n^{th}$ term of a sequence.

For example, the sequence 2, 4, 6, 8, ... has terms:
* $u_1 = 2$ (the first term)
* $u_2 = 4$ (the second term)
* $u_3 = 6$ (the third term)
* $u_4 = 8$ (the fourth term)

Sequences can go on forever (infinite) or have a specific end (finite).

What is a Series?

A series is the sum of the terms in a sequence. If you have a sequence, adding up its terms gives you a series. We use $S_n$ to denote the sum of the first $n$ terms.

For the sequence 2, 4, 6, 8, ...
* The first term is 2.
* The sum of the first two terms ($S_2$) is $2 + 4 = 6$.
* The sum of the first three terms ($S_3$) is $2 + 4 + 6 = 12$.

How are Terms Defined?

Terms in a sequence are often defined by a rule or a formula.

1. General Term Formula (Explicit Formula): This formula lets you find any term $u_n$ directly by knowing its position $n$.
   For the sequence 2, 4, 6, 8, ..., the general term formula is $u_n = 2n$.

   • For $n=1$, $u_1 = 2(1) = 2$.
   • For $n=4$, $u_4 = 2(4) = 8$.

2. Recurrence Relation (Recursive Formula): This formula defines a term based on one or more previous terms. You usually need to know the first term(s) to get started.
   For the sequence 2, 4, 6, 8, ..., a recurrence relation could be $u_n = u_{n-1} + 2$, with $u_1 = 2$.

   • Since $u_1 = 2$, $u_2 = u_1 + 2 = 2 + 2 = 4$.
   • Then $u_3 = u_2 + 2 = 4 + 2 = 6$.

Here's how these concepts link together:

graph TD
    A["Sequence (Ordered List)"] --> B["Terms ($u_n$)"]
    B --> C["General Term Formula ($u_n = f(n)$)"]
    B --> D["Recurrence Relation ($u_n = u_{n-1} + d$ or $u_n = r \cdot u_{n-1}$)"]
    A --> E["Series (Sum of Terms)"]
    E --> F["Sum of First n Terms ($S_n$)"]
    F --> G["Formula for Summing specific Series"]

Types of Sequences: Arithmetic and Geometric Progressions

These are two common and important types of sequences:

Arithmetic Progression (AP)

In an Arithmetic Progression, the difference between consecutive terms is constant. This constant difference is called the common difference, usually denoted by $d$.

• Example: 3, 7, 11, 15, ...
  • $u_2 - u_1 = 7 - 3 = 4$
  • $u_3 - u_2 = 11 - 7 = 4$
  • The common difference $d=4$.
• General term formula: $u_n = u_1 + (n-1)d$
• Sum of the first $n$ terms ($S_n$): $S_n = \frac{n}{2}(u_1 + u_n)$ or $S_n = \frac{n}{2}(2u_1 + (n-1)d)$

Geometric Progression (GP)

In a Geometric Progression, the ratio between consecutive terms is constant. This constant ratio is called the common ratio, usually denoted by $r$.

• Example: 2, 6, 18, 54, ...
  • $u_2 / u_1 = 6 / 2 = 3$
  • $u_3 / u_2 = 18 / 6 = 3$
  • The common ratio $r=3$.
• General term formula: $u_n = u_1 r^{n-1}$
• Sum of the first $n$ terms ($S_n$): $S_n = \frac{u_1(r^n - 1)}{r - 1}$ (for $r \neq 1$)
• For an infinite GP to have a finite sum, the common ratio $r$ must satisfy $|r| < 1$. The sum to infinity is $S_\infty = \frac{u_1}{1 - r}$.

3. Worked Example

Let's look at the sequence 5, 8, 11, 14, ...

1. Identify the type of sequence:

   • Check differences: $8-5=3$, $11-8=3$, $14-11=3$. The common difference is $d=3$. This is an Arithmetic Progression.
   • The first term $u_1 = 5$.

2. Find the general term formula ($u_n$):

   • Using the AP formula: $u_n = u_1 + (n-1)d$
   • Substitute $u_1=5$ and $d=3$: $u_n = 5 + (n-1)3$
   • Simplify: $u_n = 5 + 3n - 3 \Rightarrow u_n = 3n + 2$.

3. Find the 10th term ($u_{10}$):

   • Using the general term formula: $u_{10} = 3(10) + 2 = 30 + 2 = 32$.

4. Find the sum of the first 10 terms ($S_{10}$):

   • Using the AP sum formula: $S_n = \frac{n}{2}(u_1 + u_n)$
   • We know $n=10$, $u_1=5$, and we just found $u_{10}=32$.
   • $S_{10} = \frac{10}{2}(5 + 32) = 5(37) = 185$.

4. Key Takeaways

• A sequence is an ordered list of numbers ($u_1, u_2, u_3, ...$).
• A series is the sum of the terms in a sequence ($S_n = u_1 + u_2 + ... + u_n$).
• Sequences can be defined by a general term formula ($u_n = f(n)$) or a recurrence relation ($u_n$ based on previous terms).
• An Arithmetic Progression (AP) has a constant common difference ($d$) between terms.
• A Geometric Progression (GP) has a constant common ratio ($r$) between terms.
• Remember the formulas for the $n^{th}$ term and the sum of the first $n$ terms for both AP and GP.
• An infinite GP only has a finite sum if the absolute value of its common ratio $|r|$ is less than 1.

Common mistakes to avoid:
- Confusing the general term ($u_n$) with the sum ($S_n$).
- Mixing up AP and GP formulas (e.g., using $d$ in a GP formula or $r$ in an AP formula).
- Incorrectly applying the order of operations when calculating terms or sums, especially with exponents in GP.
- Forgetting that the sum to infinity for a GP only works when $|r| < 1$.

5. Now Try It

Consider the sequence 4, 12, 36, ...

1. Determine if this is an Arithmetic or Geometric Progression, and state its first term ($u_1$) and common difference ($d$) or common ratio ($r$).
2. Write down the general term formula for this sequence.
3. Calculate the 5th term of the sequence ($u_5$).
4. Calculate the sum of the first 5 terms of the sequence ($S_5$).

Success looks like: You correctly identify the type of sequence, provide the correct formula, and accurately calculate the 5th term and the sum of the first 5 terms.