Foundations of Sequences

TL;DR

A sequence is an ordered list of numbers, and each number in the list is called a term. We often look for a pattern or rule to predict the next term in a sequence. Understanding sequences helps you see how things change over time in a predictable way.

1. The Mental Model

Think of a sequence like a train of numbers, where each car (term) follows the one before it in a specific order. Each car might have a number plate that tells you its position, and there's often a hidden instruction telling you how to get from one car to the next.

2. The Core Material

A sequence is simply an ordered list of numbers. Each number in the sequence is called a term. We often denote the terms of a sequence using a letter with a subscript, like $a_n$, where 'n' tells you its position.

• $a_1$ is the first term.
• $a_2$ is the second term.
• $a_n$ is the $n$-th term (or the term in the $n$-th position).

The '...' at the end of a sequence means it continues infinitely. If it stops, it's a finite sequence.

Identifying Patterns

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The main goal when you're looking at a sequence is to figure out the rule or pattern that generates it. This rule can tell you how to get from one term to the next (a recursive rule) or how to find any term directly from its position (an explicit rule).

Common Pattern Types:

1. Adding/Subtracting a constant: Each term is found by adding or subtracting the same number from the previous term. This is an arithmetic sequence.
   • Example: 2, 5, 8, 11, ... (add 3 each time)
2. Multiplying/Dividing by a constant: Each term is found by multiplying or dividing the previous term by the same number. This is a geometric sequence.
   • Example: 3, 6, 12, 24, ... (multiply by 2 each time)
3. Other patterns: Sometimes the rule involves squaring numbers, adding increasing amounts, or combining operations.

Recursive vs. Explicit Rules

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• Recursive Rule: This rule tells you how to find a term using the previous term(s). You need to know where to start (the first term).
  • Example: For the sequence 2, 5, 8, 11, ... the recursive rule is $a_n = a_{n-1} + 3$, with $a_1 = 2$. This means "the current term is the previous term plus 3".
• Explicit Rule: This rule tells you how to find any term directly, just by knowing its position ($n$). You don't need the previous terms.
  • Example: For the sequence 2, 5, 8, 11, ... the explicit rule is $a_n = 3n - 1$. If you want the 4th term ($n=4$), you calculate $3(4) - 1 = 12 - 1 = 11$.

It's often easier to spot a recursive pattern first, but an explicit rule is usually more powerful for finding terms far down the line.

graph TD
    A["Start with a Sequence"] --> B{"Is there a constant difference
 between terms?"}
    B -- "Yes" --> C["Arithmetic Sequence"]
    C --> D["Recursive Rule:
 a_n = a_{n-1} + d
(d is the common difference)"]
    C --> E["Explicit Rule:
 a_n = a_1 + (n-1)d"]
    B -- "No" --> F{"Is there a constant ratio
 between terms?"}
    F -- "Yes" --> G["Geometric Sequence"]
    G --> H["Recursive Rule:
 a_n = a_{n-1} * r
(r is the common ratio)"]
    G --> I["Explicit Rule:
 a_n = a_1 * r^(n-1)"]
    F -- "No" --> J["Look for other patterns
(squares, increasing differences, etc.)"]
    J --> K["Try to find an Explicit Rule
 or a more complex Recursive Rule"]

3. Worked Example

Let's find the next three terms and both a recursive and an explicit rule for the sequence: 5, 9, 13, 17, ...

1. Identify the pattern:

   • From 5 to 9: add 4
   • From 9 to 13: add 4
   • From 13 to 17: add 4
     It looks like we're adding 4 each time. This is an arithmetic sequence with a common difference ($d$) of 4.

2. Find the next three terms:

   • The 5th term ($a_5$) would be $17 + 4 = 21$.
   • The 6th term ($a_6$) would be $21 + 4 = 25$.
   • The 7th term ($a_7$) would be $25 + 4 = 29$.
     The sequence continues: 5, 9, 13, 17, 21, 25, 29, ...

3. Write the recursive rule:

   • We know the first term ($a_1 = 5$) and the common difference ($d = 4$).
   • The recursive rule is $a_n = a_{n-1} + 4$, with $a_1 = 5$.

4. Write the explicit rule:

   • For an arithmetic sequence, the explicit rule is $a_n = a_1 + (n-1)d$.
   • Substitute $a_1 = 5$ and $d = 4$:
     $a_n = 5 + (n-1)4$
     $a_n = 5 + 4n - 4$
     $a_n = 4n + 1$
   • Let's check it:
     • For $n=1$: $a_1 = 4(1) + 1 = 5$ (Correct!)
     • For $n=2$: $a_2 = 4(2) + 1 = 9$ (Correct!)
     • For $n=4$: $a_4 = 4(4) + 1 = 17$ (Correct!)

4. Key Takeaways

• A sequence is an ordered list of numbers, and each number is a term.
• You can find the terms of a sequence by following a specific pattern or rule.
• A recursive rule tells you how to get the next term from the previous one.
• An explicit rule lets you find any term directly based on its position.
• Arithmetic sequences have a constant difference between consecutive terms.
• Geometric sequences have a constant ratio (multiplier) between consecutive terms.

Common Mistakes to Avoid:

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• Don't assume a pattern after only one or two differences/ratios; check at least three terms.
• Be careful with the starting term ($a_1$) when creating rules; it's crucial for both recursive and explicit forms.
• Forgetting to include the starting term ($a_1$) when giving a recursive rule makes it incomplete.
• Mixing up arithmetic (adding/subtracting) and geometric (multiplying/dividing) rules.

5. Now Try It

Given the sequence: 1, 3, 9, 27, ...

1. Identify the type of sequence.
2. Find the next three terms.
3. Write both a recursive rule and an explicit rule for this sequence.

You've successfully completed this exercise if you've correctly identified it as a geometric sequence, found the next three terms (81, 243, 729), provided a recursive rule like $a_n = a_{n-1} \times 3$ with $a_1=1$, and an explicit rule like $a_n = 1 \times 3^{n-1}$.