Introduction to Recursive Rules for Sequences
Grade 9 Algebra 1 (California Reveal Math, Unit 8: Exponential Functions) introduces recursive formulas for sequences. A recursive formula has two parts: an initial condition giving the first term a1, and a recursive rule expressing each term using the previous one as a_n = f(a_{n-1}). For example, the sequence 3, 7, 11, 15 has a1=3 and a_n = a_{n-1} + 4, while 2, 6, 18, 54 has a1=2 and a_n = 3 * a_{n-1}. Unlike explicit formulas, recursive rules require knowing the previous term to find the next.
Key Concepts
A recursive formula defines each term of a sequence using one or more previous terms. For single step sequences, it has two parts:.
1. The initial condition : the value of the first term, $a 1$ 2. The recursive rule : a formula expressing $a n$ in terms of the immediately preceding term $a {n 1}$.
Common Questions
What are the two required parts of a recursive formula?
Every recursive formula needs an initial condition (the value of the first term a1) and a recursive rule (a formula expressing a_n in terms of a_{n-1} for n >= 2).
What is the recursive formula for the sequence 2, 6, 18, 54?
a1 = 2 and a_n = 3 * a_{n-1}. Each term is found by multiplying the previous term by 3. This is a geometric sequence with ratio r = 3.
How do you find the 4th term using a recursive formula?
For a1 = 5 and a_n = a_{n-1} - 2: compute step by step. a2 = 3, a3 = 1, a4 = -1. You must calculate each term in order.
How does a recursive formula differ from an explicit formula?
An explicit formula gives any term directly by position n. A recursive formula requires you to know the previous term before finding the next one, so you must calculate sequentially.
Does a recursive formula work for both arithmetic and geometric sequences?
Yes. Arithmetic sequences use a_n = a_{n-1} + d (adding constant d), and geometric sequences use a_n = r * a_{n-1} (multiplying by constant ratio r). Both are single-step recursive rules.