Arithmetic Number Patterns (Linear Sequences)

TL;DR

Arithmetic number patterns, also called linear sequences, are lists of numbers where you add or subtract the same value to get from one term to the next. This consistent change is called the common difference. You can use a simple formula to find any term in the sequence or to create the formula for the sequence itself.

1. The Mental Model

Imagine a staircase where every step is exactly the same height. If you know the height of the first step and the height of each subsequent step, you can figure out the height of any step without measuring them all. That's essentially an arithmetic pattern.

2. The Core Material

An arithmetic number pattern (or linear sequence) is just a list of numbers where the difference between consecutive terms is constant. This constant difference is super important; it's called the common difference.

Finding the Common Difference

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To find the common difference, pick any term and subtract the term directly before it.

• Example: In the sequence 2, 5, 8, 11, 14...

  • $5 - 2 = 3$
  • $8 - 5 = 3$
  • $11 - 8 = 3$
  • The common difference is 3.

• Example: In the sequence 20, 15, 10, 5...

  • $15 - 20 = -5$
  • $10 - 15 = -5$
  • The common difference is -5 (it's decreasing!).

The General Term Formula

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You can find any term in an arithmetic sequence using a specific formula. It's often called the 'nth term' formula:

$a_n = a_1 + (n-1)d$

Let's break that down:
* $a_n$ is the term you want to find (e.g., the 10th term, the 50th term).
* $a_1$ is the very first term in the sequence.
* $n$ is the position of the term you want (e.g., if you want the 10th term, $n=10$).
* $d$ is the common difference.

This formula essentially says: start at the first term ($a_1$), then add the common difference ($d$) a total of $(n-1)$ times to reach the $n$-th term. Why $(n-1)$? Because you don't add the difference to get to the first term; you add it between terms. To get to the second term, you add 'd' once ($2-1=1$). To get to the third term, you add 'd' twice ($3-1=2$).

Steps to Find the Formula for a Sequence

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It's common to be given a sequence and asked to find its general formula.

graph TD
    A["Start with sequence: e.g., 3, 7, 11, 15..."] --> B["Identify 'a1' (first term)"];
    B --> C["Calculate 'd' (common difference)"];
    C --> D["Substitute 'a1' and 'd' into: a_n = a_1 + (n-1)d"];
    D --> E["Simplify the expression"];
    E --> F["Check with a term (e.g., n=2)"];
    F --> G["Final Formula"];

Let's apply these steps:

Sequence: 3, 7, 11, 15...

1. Identify $a_1$ (first term): $a_1 = 3$
2. Calculate $d$ (common difference): $7 - 3 = 4$. So, $d = 4$.
3. Substitute into $a_n = a_1 + (n-1)d$:
   $a_n = 3 + (n-1)4$
4. Simplify the expression:
   $a_n = 3 + 4n - 4$
   $a_n = 4n - 1$
5. Check with a term (e.g., n=2):
   Using our formula: $a_2 = 4(2) - 1 = 8 - 1 = 7$.
   Looking at the original sequence, the second term is 7. It works!
6. Final Formula: $a_n = 4n - 1$

Now you have a formula that generates any term in that sequence!

3. Worked Example

Let's say you're given the arithmetic sequence: 10, 7, 4, 1, ...
And you need to find the 20th term ($a_{20}$).

1. Identify the first term ($a_1$):
   The first term is 10. So, $a_1 = 10$.

2. Find the common difference ($d$):
   Subtract any term from the one after it.
   $7 - 10 = -3$
   $4 - 7 = -3$
   So, the common difference $d = -3$.

3. Use the general term formula $a_n = a_1 + (n-1)d$:
   You want the 20th term, so $n = 20$.
   Substitute the values you found:
   $a_{20} = 10 + (20-1)(-3)$

4. Calculate:
   $a_{20} = 10 + (19)(-3)$
   $a_{20} = 10 - 57$
   $a_{20} = -47$

The 20th term of the sequence 10, 7, 4, 1, ... is -47.

4. Key Takeaways

• Arithmetic sequences have a constant difference between consecutive terms.
• This constant difference is called the common difference ($d$).
• The formula $a_n = a_1 + (n-1)d$ lets you find any term in an arithmetic sequence.
• $a_1$ is the first term, $n$ is the term's position, and $d$ is the common difference.
• To find the general formula for a sequence, first find $a_1$ and $d$, then substitute and simplify.

Common Mistakes to Avoid:
- Forgetting that $(n-1)$ multiplies the common difference, not just $n$.
- Not paying attention to negative common differences when the sequence is decreasing.
- Confusing the term number ($n$) with the value of the term ($a_n$).
- Making calculation errors when simplifying the general formula.

5. Now Try It

You're given the sequence: 5, 12, 19, 26, ...

Your task is to:
1. Find the common difference ($d$).
2. Write the general formula for this sequence (in the form $a_n = \text{something}$).
3. Use your formula to find the 15th term of the sequence.

Success looks like: You have identified $d$, written a correct general formula, and calculated the 15th term accurately using your formula.