What is a recursive formula?
A recursive formula for a sequence defines each term using the previous term(s).
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Instead of giving a formula for the nth term directly, it tells you how to get the next term from the current one.
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Recursive formulas are very useful for arithmetic, geometric, and mixed sequences.
Notation:
an=expression involving an−1a_n = \text{expression involving } a_{n-1}
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a_n= current term -
a_{n-1}= previous term -
Must also specify the first term
a₁
Step 1: Recursive formula for an arithmetic sequence
Example:
Arithmetic sequence: 3, 7, 11, 15, …
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Common difference:
d = 4 -
Recursive formula:
{a1=3an=an−1+4,n≥2\begin{cases} a_1 = 3 \\ a_n = a_{n-1} + 4, \quad n \ge 2 \end{cases}
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Meaning: start at 3, then add 4 each time
✅ Check:
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a₂ = 3 + 4 = 7
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a₃ = 7 + 4 = 11
Step 2: Recursive formula for a geometric sequence
Example:
Geometric sequence: 2, 6, 18, 54, …
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Common ratio:
r = 3 -
Recursive formula:
{a1=2an=3⋅an−1,n≥2\begin{cases} a_1 = 2 \\ a_n = 3 \cdot a_{n-1}, \quad n \ge 2 \end{cases}
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Meaning: start at 2, then multiply by 3 each time
✅ Check:
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a₂ = 2 × 3 = 6
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a₃ = 6 × 3 = 18
Step 3: Recursive formula for a mixed sequence
Example:
Sequence: 2, 4, 5, 10, 11, 22, …
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Pattern: multiply by 2, then add 1
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Recursive formula (depending on n):
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Or you can alternate operations carefully
✅ Next term check:
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2 → 4 (×2)
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4 → 5 (+1)
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5 → 10 (×2)
Step 4: How to use a recursive formula
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Start with the first term a₁
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Apply the recursive rule repeatedly to find a₂, a₃, …
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Continue until you reach the term you need
Example:
Arithmetic recursive formula:
a1=5,an=an−1+7a_1 = 5, \quad a_n = a_{n-1} + 7
Find first 5 terms:
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a₁ = 5
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a₂ = 5 + 7 = 12
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a₃ = 12 + 7 = 19
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a₄ = 19 + 7 = 26
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a₅ = 26 + 7 = 33
✅ Sequence: 5, 12, 19, 26, 33
Step 5: Key points to remember
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Always specify the first term
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Recursive formulas tell how to go from one term to the next
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They are different from explicit formulas, which give any term directly
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Useful for complex sequences or sequences where the nth term formula is hard to find
Summary
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Recursive formula = each term depends on previous term(s)
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Arithmetic:
a_n = a_{n-1} + d -
Geometric:
a_n = r * a_{n-1} -
Mixed: operations may alternate, adjust formula accordingly
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Always include the first term a₁