Recursive Formula for Geometric Sequences (Property Form)
Recursive formulas for geometric sequences show Grade 11 Algebra 1 students how each term is found by multiplying the previous term by the common ratio r. Covered in enVision Chapter 6, the formula a_n = r · a_(n-1) requires two pieces of information: the first term and the common ratio. For instance, the sequence 3, 6, 12, 24 uses a_n = 2 · a_(n-1) with a_1 = 3, while 80, 40, 20, 10 uses a_n = 0.5 · a_(n-1). Unlike explicit formulas, recursive formulas require computing each prior term in sequence to reach a target term.
Key Concepts
A recursive formula for a geometric sequence expresses each term in relation to the previous term: $$a n = r \cdot a {n 1}$$ where $r$ is the common ratio and $a {n 1}$ is the previous term.
Common Questions
What is the recursive formula for a geometric sequence?
The recursive formula is a_n = r · a_(n-1), where r is the common ratio and a_(n-1) is the previous term. You also need to know the first term a_1.
How do you find the common ratio for a recursive geometric formula?
Divide any term by the term before it. For the sequence 3, 6, 12, 24, dividing 6 ÷ 3 = 2 gives the common ratio r = 2.
What is the recursive formula for 80, 40, 20, 10?
The formula is a_n = 0.5 · a_(n-1) with a_1 = 80, since each term is half the previous term.
How is a recursive formula different from an explicit formula?
A recursive formula defines each term using the previous term, so you must compute all terms in order. An explicit formula lets you find any term directly using its position number.
Can a geometric sequence have a ratio less than 1?
Yes. A ratio between 0 and 1 produces a decreasing sequence. For example, r = 0.5 gives the sequence 80, 40, 20, 10.
Why do you need the first term for a recursive formula?
The formula only tells you how each term relates to the previous one. Without the starting value a_1, you cannot determine any specific term in the sequence.