Practice Recursive vs Explicit Formulas in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick 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.

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}.

Example 2

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

Example 3

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

Example 4

medium
Convert a_n = 4n - 1 to a recursive formula.
← Back to Recursive vs Explicit Formulas Worked Examples Formula Explained