Recursive Definition of Arithmetic Sequences
Recursive definition of arithmetic sequences is a Grade 11 algebra concept in Big Ideas Math. An arithmetic sequence has a constant common difference d between consecutive terms. A recursive formula defines each term using the previous term: aₙ = aₙ₋₁ + d, with a given starting term a₁. For example, the sequence 3, 7, 11, 15,… has d = 4 and is defined recursively as a₁ = 3, aₙ = aₙ₋₁ + 4. Unlike the explicit formula aₙ = a₁ + (n−1)d, the recursive formula requires knowing the previous term to find the next. Both forms describe the same sequence; recursive is natural for computing terms step-by-step.
Key Concepts
The recursive definition of an arithmetic sequence is written as: $$a n = \begin{cases} a 1, & \text{if } n = 1 \\ a {n 1} + d, & \text{if } n 1 \end{cases}$$ where $a 1$ is the first term and $d$ is the common difference.
Common Questions
What is a recursive formula for an arithmetic sequence?
A recursive formula defines each term using the previous term plus the common difference: aₙ = aₙ₋₁ + d. A starting value a₁ must also be given.
What is the recursive formula for the sequence 3, 7, 11, 15,…?
The common difference is d = 4. Recursive form: a₁ = 3, aₙ = aₙ₋₁ + 4 for n ≥ 2.
What is the difference between a recursive and explicit formula for arithmetic sequences?
The explicit formula aₙ = a₁ + (n−1)d calculates any term directly from n. The recursive formula requires knowing the previous term; it is computed step-by-step.
How do you find the 5th term using a recursive formula if a₁ = 3 and d = 4?
a₂ = 3+4=7, a₃ = 7+4=11, a₄ = 11+4=15, a₅ = 15+4=19. Each step uses the previous result.
How do you identify the common difference in an arithmetic sequence?
Subtract any term from the next term: d = aₙ − aₙ₋₁. The difference must be constant throughout the sequence.
When is a recursive formula more useful than an explicit formula?
Recursive formulas are natural for computer algorithms and situations where terms are generated step-by-step. Explicit formulas are faster for finding a specific far-off term directly.