What is an arithmetic sequence and how do I solve problems involving them?
What is an Arithmetic Sequence?
An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant value is known as the common difference (denoted as d). If you know the starting number and the amount by which the sequence grows or shrinks, you can predict any future term in the series.
The Anatomy of a Sequence
To master these sequences, you need to identify three core components:
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First Term ($a_1$): The very first number in your list.
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Common Difference ($d$): The amount added to each term to get the next one. This can be positive (increasing) or negative (decreasing).
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The $n^{th}$ Term ($a_n$): The specific value located at position n in the sequence.
The Golden Formula
To find any term without writing out the entire list, use this formula:
$a_n = a_1 + (n - 1)d$
This formula works because to get to the $n^{th}$ term, you start at the first term and add the common difference $(n-1)$ times.
Quick Reference Table
| Feature | Description | Example ($2, 5, 8, 11...$) |
|---|---|---|
| First Term ($a_1$) | The starting point | $2$ |
| Common Difference ($d$) | The step size | $3$ |
| $n^{th}$ Term Formula | The general rule | $a_n = 2 + (n-1)3$ |
Real-World Examples
Arithmetic sequences are everywhere in daily life. Consider these scenarios:
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Savings Goals: If you deposit $50 into a savings account every month, your total balance forms an arithmetic sequence where $d = 50$.
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Theater Seating: If the first row has 10 seats and every subsequent row adds 2 seats, the number of seats per row follows an arithmetic sequence.
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Fitness Progress: If you start by running 1 mile and add 0.5 miles to your distance each week, your training plan is an arithmetic sequence.
Common Pitfalls
Even experts make mistakes when working with sequences. Watch out for these common traps:
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The "Off-by-One" Error: Students often forget the $(n-1)$ part of the formula. Remember: you don't add the difference to the first term to get the first term; you only start adding it from the second term onward.
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Confusing $n$ and $a_n$: Remember that $n$ is the position (e.g., 1st, 2nd, 3rd), while $a_n$ is the value at that position.
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Negative Differences: If the sequence is decreasing (e.g., $10, 8, 6$), the common difference is negative ($d = -2$). Always double-check your signs!