Recursive vs Explicit Formulas Math Example 1

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

Example 1

easy
A sequence is defined by a1=3a_1 = 3, an=an−1+5a_n = a_{n-1} + 5. Find an explicit formula and compute a50a_{50}.

Solution

  1. 1
    First few terms: 3,8,13,18,…3, 8, 13, 18, \ldots — arithmetic with d=5d=5.
  2. 2
    Explicit: an=3+(n−1)⋅5=5n−2a_n = 3 + (n-1) \cdot 5 = 5n - 2.
  3. 3
    Verify: a1=3a_1 = 3 ✓, a2=8a_2 = 8 ✓.
  4. 4
    a50=5(50)−2=248a_{50} = 5(50)-2 = 248.

Answer

an=5n−2a_n = 5n-2; a50=248a_{50} = 248
A constant-difference recursion defines an arithmetic sequence. The explicit formula an=a1+(n−1)da_n = a_1 + (n-1)d lets you reach any term instantly.

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 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]

Example 5 medium

Convert [formula] to a recursive formula.

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