Recursive Formula for Arithmetic Sequences
A recursive formula for an arithmetic sequence defines each term using the previous term: a₁ = first term and aₙ = aₙ₋₁ + d, where d is the common difference — a key concept in enVision Algebra 1 Chapter 3 for Grade 11. For the sequence 3, 7, 11, 15, ..., the recursive formula is a₁ = 3 and aₙ = aₙ₋₁ + 4. For 12, 9, 6, 3, ..., the difference d = -3, giving a₁ = 12 and aₙ = aₙ₋₁ + (-3). Unlike an explicit formula, a recursive formula requires knowing the previous term to find the next, making it useful for describing patterns step-by-step.
Key Concepts
A recursive formula for an arithmetic sequence defines each term using the previous term: $a 1 = \text{first term}$ and $a n = a {n 1} + d$ where $d$ is the common difference.
Common Questions
What is a recursive formula for an arithmetic sequence?
A recursive formula has two parts: the first term a₁, and a rule showing how each term relates to the previous one: aₙ = aₙ₋₁ + d, where d is the common difference.
What is the recursive formula for 3, 7, 11, 15, ...?
The common difference is 7 - 3 = 4, so the formula is a₁ = 3 and aₙ = aₙ₋₁ + 4.
What is the recursive formula for 12, 9, 6, 3, ...?
The common difference is 9 - 12 = -3, so the formula is a₁ = 12 and aₙ = aₙ₋₁ + (-3) or aₙ = aₙ₋₁ - 3.
How do you find the common difference d for a recursive formula?
Subtract any term from the next: d = a₂ - a₁. The common difference is constant throughout the arithmetic sequence.
How does a recursive formula differ from an explicit formula?
A recursive formula requires the previous term to find the next; for example you need a₄ to find a₅. An explicit formula like aₙ = a₁ + (n-1)d lets you find any term directly without knowing the preceding ones.