In Algebra 1 you described Arithmetic Sequences and Geometric Sequences with explicit formulas, which jump straight to any term. A recursive formula describes the same sequences step by step instead: where to start, and how to get from one term to the next.
Key idea
A recursive formula gives (1) the first term, such as $a_1 = 5$, and (2) a rule for each later term from the one before it, such as $a_n = a_{n-1} + 3$ for $n \ge 2$. Arithmetic: $a_n = a_{n-1} + d$. Geometric: $a_n = r \cdot a_{n-1}$.
Why it works
Many processes are naturally described one step at a time: a savings account that earns interest and gets a deposit each month, $a_n = 1.01\,a_{n-1} + 50$, or a population that doubles every year. The recursive rule says exactly what happens in one step. The explicit formula is what you get after following the rule $n - 1$ times: adding $d$ that many times gives $a_1 + (n - 1)d$.
Worked examples
Worked example: Following a rule
$a_1 = 2$ and $a_n = 3a_{n-1} - 1$. Find $a_4$.
- $a_2 = 3(2) - 1 = 5$.
- $a_3 = 3(5) - 1 = 14$.
- $a_4 = 3(14) - 1 = 41$.
Answer: $a_4 = 41$
Worked example: From explicit to recursive
Write a recursive formula for $a_n = 4n + 1$.
- First term: $a_1 = 4(1) + 1 = 5$.
- Each time $n$ goes up by 1, $a_n$ goes up by 4.
Answer: $a_1 = 5$, $a_n = a_{n-1} + 4$ for $n \ge 2$
Worked example: Two previous terms
$a_1 = 1$, $a_2 = 1$ and $a_n = a_{n-1} + a_{n-2}$. List the first 7 terms.
- Each term is the sum of the two before it: $1, 1, 2, 3, 5, 8, 13$.
Answer: $1, 1, 2, 3, 5, 8, 13$ (the Fibonacci numbers)
Common mistake
Mixing up $a_n$ and $a_{n+1}$. "$a_n = a_{n-1} + 3$" and "$a_{n+1} = a_n + 3$" say the same thing (each term is 3 more than the one before), but "$a_{n-1} = a_n + 3$" says each term is 3 less. Read it in words: "this term equals the previous term plus 3." Then check by computing $a_2$ from $a_1$.