AP Precalculus: Sequences – Explicit & Recursive Formulas

1. Sequence, Explicit vs. Recursive

• Sequence: Ordered list of terms \( a_1, a_2, a_3, \ldots \)
• Explicit: \( a_n \) given directly as formula in \( n \)
• Recursive: Each \( a_{n} \) based on one (or more) previous term(s)

2. Arithmetic Sequence

• Explicit: \( a_n = a_1 + (n-1)d \)
• Recursive: \( a_1 = \dots \), \( a_{n} = a_{n-1} + d \)
• \( d = \) common difference

3. Geometric Sequence

• Explicit: \( a_n = a_1 \cdot r^{n-1} \)
• Recursive: \( a_1 = \dots \), \( a_{n} = r\, a_{n-1} \)
• \( r = \) common ratio

4. Finding Terms

• Use the sequence’s formula to compute any term \( a_n \)
• Recursive: start with \( a_1 \) (or initial term), use repeatedly
• Explicit: plug in \( n \) directly

5. Convert Recursive to Explicit & Vice Versa

• Arithmetic: \( a_{n} = a_{n-1} + d \) ⇒ \( a_{n} = a_1 + (n-1)d \)
• Geometric: \( a_{n} = r a_{n-1} \) ⇒ \( a_{n} = a_1 r^{n-1} \)
• To create a recursive from explicit: solve for \( a_n \) in terms of \( a_{n-1} \), define \( a_1 \)

6. Notation & Mixed Examples

• Use \( a_n \) or \( s_n \) for “nth term”
• Example explicit: \( a_n = 7 + 5(n-1) \) (arithmetic, \( d=5 \))
• Example recursive: \( s_{n}=3s_{n-1} \), \( s_1 = 2 \) (geometric, \( r=3 \))