Recursive vs Explicit Formulas Math Example 2

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Example 2

medium
A sequence satisfies a1=2a_1 = 2, an=3an−1a_n = 3a_{n-1}. Find the explicit formula and identify the growth type.

Solution

  1. 1
    Terms: 2,6,18,54,…2, 6, 18, 54, \ldots — each multiplied by 3.
  2. 2
    Geometric with r=3r=3: explicit formula an=2⋅3n−1a_n = 2 \cdot 3^{n-1}.
  3. 3
    Check: a1=2⋅1=2a_1 = 2 \cdot 1 = 2 ✓; a2=2⋅3=6a_2 = 2 \cdot 3 = 6 ✓.
  4. 4
    This represents exponential growth.

Answer

an=2⋅3n−1a_n = 2 \cdot 3^{n-1} (exponential growth)
A constant-ratio recursion defines a geometric sequence. The explicit formula is exponential in nn, far more efficient to evaluate than iterating the recursion.

About Recursive vs Explicit Formulas

Two ways to define a sequence: recursive uses the previous term(s), explicit gives the nnth term directly as a function of nn.

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More Recursive vs Explicit Formulas Examples

Example 1 easy

A sequence is defined by [formula], [formula]. Find an explicit formula and compute [formula].

Example 3 medium

A sequence is defined by [formula], [formula]. Write the explicit formula and find [formula].

Example 4 easy

Write a recursive formula for [formula]

Example 5 medium

Convert [formula] to a recursive formula.

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