Recursive vs Explicit Formulas Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Recursive vs Explicit Formulas.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Two ways to define a sequence: a recursive formula defines each term using the previous term(s), while an explicit (closed-form) formula gives the nth term directly as a function of n.

A recursive formula is like step-by-step directions ('from where you are, go 3 blocks north'). An explicit formula is like GPS coordinates ('go to 5th Avenue and 42nd Street'). Both describe the same sequence, but explicit formulas let you jump to any term instantly.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Recursive formulas describe the pattern locally (term-to-term), while explicit formulas describe it globally (position-to-value). Converting from recursive to explicit reveals the underlying structure of the sequence.

Common stuck point: To convert recursive to explicit, look for patterns: constant difference means arithmetic (linear), constant ratio means geometric (exponential). For more complex recurrences (like Fibonacci), the techniques are more advanced.

Sense of Study hint: Build a small table of n and a_n values using the recursive rule, then look for a linear or exponential pattern.

Worked Examples

Example 1

easy
A sequence is defined by a_1 = 3, a_n = a_{n-1} + 5. Find an explicit formula and compute a_{50}.

Solution

  1. 1
    First few terms: 3, 8, 13, 18, \ldots — arithmetic with d=5.
  2. 2
    Explicit: a_n = 3 + (n-1) \cdot 5 = 5n - 2.
  3. 3
    Verify: a_1 = 3 ✓, a_2 = 8 ✓.
  4. 4
    a_{50} = 5(50)-2 = 248.

Answer

a_n = 5n-2; a_{50} = 248
A constant-difference recursion defines an arithmetic sequence. The explicit formula a_n = a_1 + (n-1)d lets you reach any term instantly.

Example 2

medium
A sequence satisfies a_1 = 2, a_n = 3a_{n-1}. Find the explicit formula and identify the growth type.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Write a recursive formula for 5, 10, 20, 40, 80, \ldots

Example 2

medium
Convert a_n = 4n - 1 to a recursive formula.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

sequencearithmetic sequencegeometric sequence