Predicting What Comes Next: Exploring Sequences and Progressions · Lesson 3 of 8
Recursive Rules for Sequences
“A recursive rule says, “Ask the previous term”—mathematical word-of-mouth.”
• Understand recursive rules. • Use earlier terms to generate later terms. • Compare recursive and explicit rules. • Work with rules depending on one or more previous terms. • Understand the Virahānka–Fibonacci sequence.
A sequence can be described in different ways. An explicit rule gives us a direct formula for finding any term from its position in the sequence. This means that if we want to find, for example, the 20th term, we can calculate it directly without first finding all the terms that come before it.
A recursive rule works differently. Instead of giving the value of a term directly from its position, it explains how to find each new term using one or more earlier terms. To use a recursive rule, we usually need to know the starting term or terms first. We then follow the rule step by step to generate the rest of the sequence.
A rule that defines a term using the value of one or more previous terms.
Problem
Describe 1,4,7,10,13,… recursively.
- 1.The first term is t₁=1.
- 2.Every new term is 3 more than the previous term.
- 3.So tₙ=tₙ₋₁+3 for n≥2.
A recursive rule needs a starting value. Without t₁=1, the instruction 'add 3 to the previous term' does not tell us where the sequence begins.
Problem
Find the first four terms if u₁=1 and uₙ=2uₙ₋₁+3.
- 1.u₁=1.
- 2.u₂=2(1)+3=5.
- 3.u₃=2(5)+3=13.
- 4.u₄=2(13)+3=29.
- 5.So the sequence begins 1,5,13,29,… .
Explicit vs Recursive
| Explicit rule | Recursive rule |
|---|---|
| Uses the position n directly | Uses earlier terms |
| Can jump straight to a distant term | Usually generates terms step by step |
| Example: tₙ=3n−2 | Example: t₁=1, tₙ=tₙ₋₁+3 |
A Rule Can Use Two Previous Terms
A recursive formula is not limited to the immediately previous term. The Virahānka–Fibonacci sequence begins 1,2 and then each new term is the sum of the previous two.
Problem
Generate the first eight terms.
- 1.V₁=1 and V₂=2.
- 2.V₃=2+1=3.
- 3.V₄=3+2=5.
- 4.V₅=5+3=8.
- 5.Continuing gives 13,21,34.
- 6.So the sequence is 1,2,3,5,8,13,21,34,… .
This sequence was studied in India by Virahānka in connection with poetic metre and was later studied by Gopāla and Hemachandra, before appearing in the work of Fibonacci.
Practice Problems
- Generate the first six terms from t₁=−4, tₙ=tₙ₋₁+5.
- Find the first five terms if u₁=2 and uₙ=3uₙ₋₁−1.
- Write a recursive rule for 7,11,15,19,… .
- Write the next four terms of 1,2,3,5,8,13,… .
- Explain one advantage of an explicit rule and one advantage of a recursive rule.
- A sequence starts 2,4,7 and each new term is the sum of the previous three. Find the next five terms.
Key Takeaways
• Recursive rules use earlier terms. • A starting term or starting terms are essential. • Explicit rules jump directly to tₙ; recursive rules build step by step. • Recursive rules may depend on one, two or more earlier terms. • The Virahānka–Fibonacci sequence uses the previous two terms.
Next, we study one of the most important special sequences: arithmetic progressions, where the same amount is added each time.