Converting Explicit to Recursive Rules (Arithmetic & Geometric)
Converting explicit to recursive rules for arithmetic and geometric sequences is a Grade 9 skill in California Reveal Math (Unit 8: Exponential Functions). For arithmetic sequences with explicit formula a_n = a_1 + (n-1)d, the recursive form is a_n = a_{n-1} + d with initial condition a_1. For geometric sequences with a_n = a_1 * r^(n-1), the recursive form is a_n = r * a_{n-1}. Example: a_n = 81*(1/3)^(n-1) becomes a_1 = 81, a_n = (1/3)*a_{n-1}.
Key Concepts
To convert an explicit formula to a recursive formula , identify whether the sequence is arithmetic or geometric, then express each term in relation to the previous term.
Arithmetic (explicit: $a n = a 1 + (n 1)d$):.
Common Questions
How do you convert an arithmetic explicit formula to recursive form?
Identify a_1 and d from the explicit formula a_n = a_1 + (n-1)d. The recursive form is: a_1 = given, a_n = a_{n-1} + d. Example: a_n = 3 + (n-1)(5) becomes a_1 = 3, a_n = a_{n-1} + 5.
How do you convert a geometric explicit formula to recursive form?
Identify a_1 and r from a_n = a_1 * r^(n-1). The recursive form is: a_1 = given, a_n = r * a_{n-1}. Example: a_n = 4 * 2^(n-1) becomes a_1 = 4, a_n = 2 * a_{n-1}.
How do you convert a_n = 81*(1/3)^(n-1) to recursive form?
a_1 = 81 and r = 1/3. Recursive form: a_1 = 81, a_n = (1/3) * a_{n-1}. Each term is found by multiplying the previous by 1/3.
Must you always include the initial condition in a recursive formula?
Yes. Without the initial condition a_1, the recursive rule alone is incomplete. You cannot compute any term without knowing where the sequence starts.
What determines whether a sequence is arithmetic or geometric?
Arithmetic: constant difference d between consecutive terms. Geometric: constant ratio r between consecutive terms. Check by subtracting (arithmetic) or dividing (geometric) consecutive terms.