Recursive and Explicit Rules

Two ways to describe the same list.

12 problems · about 21 minutes · F-BF.A.2, F-IF.A.3

What this lesson teaches

The student writes recursive and explicit rules for sequences and converts between them.

A recursive rule builds each term from the previous one and needs a starting value.
An explicit rule computes any term straight from its position.
Arithmetic and geometric sequences convert easily between the two.

Straightforward practice. Get the method working first.

5 problems

aₙ = 6 + 5(n − 1). What is the 7th term?
Answer 36
Why 36.
Why must a recursive rule state a starting value?
Answer Without one, the rule describes infinitely many sequences.
Why The rule alone does not pin down the sequence.
a₁ = 2 and aₙ = aₙ₋₁ + 6. What is the 4th term?
Answer 20
Why 2, 8, 14, 20.
a₁ = 3 and aₙ = 2aₙ₋₁. What is the 4th term?
Answer 24
Why 3, 6, 12, 24.
aₙ = 4 + 3(n − 1). What is the 6th term?
Answer 19
Why 4 + 15.

The same ideas with more to keep track of.

3 problems

aₙ = 5 · 2^(n−1). What is the 4th term?
Answer 40
Why 5 × 8.
Which rule finds the 500th term without the earlier ones? 1 recursive, 2 explicit.
Answer 2
Why It uses the position directly.
Fibonacci: 1, 1, 2, 3, 5. What is the next term?
Answer 8
Why 3 + 5.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each rule with what it needs to compute a term.
Answer Recursive rule → The previous term; Explicit rule → Only the position number; Any recursive rule at all → A stated starting value
Why One of them cannot start without being told where.
aₙ = 10 + 4(n − 1). What is the 8th term?
Answer 38
Why 10 + 28.
The Recursion: a₁ = 7 and aₙ = aₙ₋₁ + 5. What is the 3rd term?
Answer 17
Why 17.
The Explicit Rule: aₙ = 7 + 5(n − 1). What is the 10th term?
Answer 52
Why 52.