AI Tutor
If you have ever watched a line of dominoes fall, you know each tile doesn't need to know where the finish line is — it only needs to react when the tile right behind it tips over.
That step-by-step chain reaction is exactly how a recursive formula works: it generates each new number using the number that came directly before it.
Every recursive formula requires two parts: the initial term (a1a_1), which acts as the starting domino, and the recursive rule, which connects the current term (ana_n) to the previous term (an−1a_{n-1}).
How do you translate a plain list of numbers into this two-part structure? Let's look at sequences where each step is an addition.

Arithmetic Sequences

In an arithmetic sequence, you add the same value — called the common difference (dd) — to get to the next term.
To find dd, take any term and subtract the term immediately preceding it: d=ak−ak−1d = a_k - a_{k-1}.
Take the sequence: 7,11,15,19,…7, 11, 15, 19, \dots. Subtracting adjacent terms gives 11−7=411 - 7 = 4 and 15−11=415 - 11 = 4, so the constant step is +4+4.
Our finished recursive formula is simply: a1=7a_1 = 7, and an=an−1+4a_n = a_{n-1} + 4 for n≥2n \ge 2.
Watch out for a frequent mistake: writing an−1a_n - 1 instead of an−1a_{n-1}. The subscript n−1{n-1} represents your address in the sequence, not a number you subtract from the term's value.
What happens when terms aren't stepping forward by addition, but multiplying and shrinking or flipping signs?