Recursive vs Explicit Formulas Math Example 5

Follow the full solution, then compare it with the other examples linked below.

Example 5

medium
Convert an=4n−1a_n = 4n - 1 to a recursive formula.

Solution

  1. 1
    a1=3a_1 = 3.
  2. 2
    an−an−1=(4n−1)−(4(n−1)−1)=4a_n - a_{n-1} = (4n-1)-(4(n-1)-1) = 4, so d=4d = 4.
  3. 3
    Recursive: a1=3a_1 = 3, an=an−1+4a_n = a_{n-1} + 4.

Answer

a1=3a_1 = 3; an=an−1+4a_n = a_{n-1} + 4
Find the first term and the constant difference between consecutive terms. A linear explicit formula always converts to a constant-difference (arithmetic) 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.

Learn more about Recursive vs Explicit Formulas →

More Recursive vs Explicit Formulas Examples

Example 1 easy

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

Example 2 medium

A sequence satisfies [formula], [formula]. Find the explicit formula and identify the growth type.

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]

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