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Recursive Formulas Practice Problems

Start from the first term and apply the rule one step at a time. The later problems convert between recursive and explicit formulas and use a two-term rule.

  1. 1.

    A sequence has and . Find .

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    6, 2, −2, −6, so .

  2. 2.

    A sequence has and . Find .

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    2, 10, 50, 250, so .

  3. 3.

    The recursive formula for 9, 13, 17, 21, … is . What is ?

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    Each term is 4 more than the one before.

  4. 4.

    The recursive formula for 80, 40, 20, 10, … is . What is ?

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    Each term is half the one before, so .

  5. 5.

    A sequence has and . Find .

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    , then .

  6. 6.

    A sequence has and . Find .

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    1, 3, 7, 15, 31, so .

  7. 7.

    Write an explicit formula for , . Type it in terms of n.

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    Arithmetic with , : .

  8. 8.

    Write an explicit formula for , . Type it in terms of n.

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    Geometric with , : .

  9. 9.

    Write an explicit formula for , . Type it in terms of n.

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    Arithmetic with : .

  10. 10.

    To write recursively you need a first term. What is ?

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    ; the rule is then .

  11. 11.

    A sequence has , and . Find .

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    2, 3, 5, 8, 13, 21, so .

  12. 12.

    A savings account starts at $200 in month 1, and $50 is added each month: , . What is the balance in month 12, in dollars?

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    Eleven deposits after month 1: .
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Recursive Formulas Practice Problems

These problems follow the recursive formulas lesson. A recursive formula needs a starting value and a rule; use both.

The two common rules

Common mistakes to watch for

  • Forgetting the starting value. A recursive formula needs both the rule and (or ).
  • Skipping a term. To find , compute in order; each depends on the one before.
  • Confusing and . is the previous term; is the current term minus one.
  • Using the wrong explicit form. Converting to explicit, an arithmetic rule gives and a geometric rule gives .

Frequently Asked Questions

How do I find a term from a recursive formula?+

Start with the given first term and apply the rule repeatedly, each time to the term you just found, until you reach the one you want.

How do I type an explicit formula?+

Use n for the position and ^ for powers, for example 5n - 3 or 3*4^(n - 1). Any equivalent form is accepted.

Calculators for this topic

Sequence Calculator

More practice problems

Arithmetic Sequences Practice Problems Geometric Sequences Practice Problems Arithmetic vs Geometric Sequences Practice Problems