AI Tutor
If someone gives you the numbers 3,6,12,…3, 6, 12, \dots, what comes next? You might say 2424 if you see doubling, or 2121 if you see differences growing by three.
A sequence is an ordered list of numbers where each number is called a term (written a1,a2,a3,…,ana_1, a_2, a_3, \dots, a_n). Identifying the exact rule that connects consecutive terms allows you to predict any future value.
What is the simplest way a sequence can grow from one term to the next?

The Constant Step: Arithmetic Sequences

An arithmetic sequence changes by adding or subtracting the exact same value between consecutive terms. This constant value is called the common difference (dd).
To test if a sequence is arithmetic, subtract each term from the one that follows it: d=an−an−1d = a_{n} - a_{n-1}
If subtraction doesn't yield a constant number, could multiplication be driving the pattern instead?

The Constant Multiplier: Geometric Sequences

A geometric sequence changes by multiplying each term by the exact same non-zero value to get the next term. This multiplier is called the common ratio (rr).
To test if a sequence is geometric, divide each term by its preceding term: r=anan−1r = \frac{a_n}{a_{n-1}}
How do you systematically check an unfamiliar list of numbers when you do not know which type it is?