Mathematics
Arithmetic Sequence
Quick fact
The ancient Babylonians recorded arithmetic sequences on clay tablets over 4,000 years ago, using them to calculate compound interest and astronomical cycles.
Why this is interesting
Have you ever noticed that the steps on a staircase rise by the same height each time? That consistent change is the essence of an arithmetic sequence, and it appears everywhere from saving money to planning city blocks.
Read the full explanation
Understanding Arithmetic Sequence
An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. Think of a staircase: each step goes up by the same amount. For example, the sequence 2, 5, 8, 11, ... starts at 2 and adds 3 each time. That constant 3 is called the common difference. To find any term, you can use the rule: start with the first term, then add the common difference as many times as needed. For instance, the 5th term is 2 + 4×3 = 14. This pattern creates a straight line when graphed, with the common difference as the slope. Arithmetic sequences model steady, predictable growth or decay—like earning $10 every week or a car losing $1,000 in value each year.
A deeper explanation
The power of an arithmetic sequence lies in its linear nature. The nth term can be expressed as an = a1 + (n-1)d, where a1 is the first term and d is the common difference. This formula is a direct application of the concept that each step adds d, so after n-1 steps you have added (n-1)d units. Graphically, the terms lie on a line with slope d and intercept a1 - d. This linear relationship connects arithmetic sequences to linear functions in algebra. Understanding this mechanism allows you to quickly calculate any term without listing all previous ones. More importantly, it reveals how constant rates of change appear in real phenomena: simple interest (adding the same amount each period), uniform motion (same distance per time), and even rhythmic patterns in music. The sum of an arithmetic sequence (arithmetic series) gives the total accumulated value, which is used in financial planning and statistical calculations. Mastering arithmetic sequences is the first step toward analyzing more complex patterns like geometric sequences or infinite series.