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Cogito

Integrated Math 1 · Chapter 5 · Lesson 3

Practice Worksheet

Recursive and Explicit Rules

Two ways to describe the same list.

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

NameDate

Figure — use these to answer the problems

An explicit rule jumps straight to any position. A recursive rule would have to walk there one point at a time.
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y = 3x + 2
Common difference3
Starting value2
Position4
Term value14
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. aₙ = 6 + 5(n − 1). What is the 7th term?

    Answer

  2. Why must a recursive rule state a starting value?

    • Without one, the rule describes infinitely many sequences.
    • It is only a writing convention.

  3. a₁ = 2 and aₙ = aₙ₋₁ + 6. What is the 4th term?

    Answer

  4. a₁ = 3 and aₙ = 2aₙ₋₁. What is the 4th term?

    Answer

  5. aₙ = 4 + 3(n − 1). What is the 6th term?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. aₙ = 5 · 2^(n−1). What is the 4th term?

    Answer

  2. Which rule finds the 500th term without the earlier ones? 1 recursive, 2 explicit.

    Answer

  3. Fibonacci: 1, 1, 2, 3, 5. What is the next term?

    Answer

Stretch Yourself

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

4 problems

  1. Match each rule with what it needs to compute a term.

    Draw a line from each item on the left to its match on the right.

    • Recursive rule
    • Explicit rule
    • Any recursive rule at all
    • The previous term
    • Only the position number
    • A stated starting value

  2. aₙ = 10 + 4(n − 1). What is the 8th term?

    Answer

  3. The Recursion

    a₁ = 7 and aₙ = aₙ₋₁ + 5. What is the 3rd term?

    Answer

  4. The Explicit Rule

    aₙ = 7 + 5(n − 1). What is the 10th term?

    Answer