Practice Recursive vs Explicit Formulas in Math

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

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Quick Recap

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

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.

Showing a random 20 of 50 problems.

Example 1

medium
Convert a1=3a_1 = 3, an=2an−1a_n = 2a_{n-1} into an explicit formula (start n=1n=1).

Example 2

easy
For the explicit formula an=(−1)na_n = (-1)^{n}, list the first five terms.

Example 3

medium
Given a1=0a_1 = 0 and an=an−1+2na_n = a_{n-1} + 2n, find a5a_5.

Example 4

medium
A sequence is defined by a1=2a_1 = 2 and an=2an−1+1a_n = 2a_{n-1} + 1. Find a closed-form explicit formula.

Example 5

hard
A sequence is defined by a1=1a_1 = 1 and an=an−1+na_n = a_{n-1} + n. Find an explicit formula.

Example 6

easy
A sequence is given by a1=5a_1 = 5 and an=an−1+3a_n = a_{n-1} + 3. Find a2a_2.

Example 7

easy
A sequence is a1=1a_1 = 1, an=an−1+na_n = a_{n-1} + n. Find a4a_4.

Example 8

easy
Write the explicit formula for the arithmetic sequence 3,7,11,15,…3, 7, 11, 15, \ldots (start at n=1n=1).

Example 9

hard
Given a1=100a_1 = 100 and an=0.9an−1+10a_n = 0.9 a_{n-1} + 10, find lim⁡n→∞an\lim_{n \to \infty} a_n.

Example 10

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

Example 11

challenge
For the recursion an=2an−1−an−2a_n = 2 a_{n-1} - a_{n-2} with a1=3a_1 = 3, a2=5a_2 = 5, find ana_n in closed form.

Example 12

medium
A sequence is defined by an=n2−na_n = n^2 - n. For which nn does an=12a_n = 12?

Example 13

medium
For an=3n+1a_n = 3n + 1, find the sum of the first 4 terms.

Example 14

easy
Which formula computes any term directly without listing earlier ones: recursive or explicit?

Example 15

hard
Solve the recurrence an=3an−1a_n = 3 a_{n-1} with a1=5a_1 = 5 for an explicit formula.

Example 16

challenge
A sequence satisfies a1=1a_1 = 1, an=an−1+2n−1a_n = a_{n-1} + 2n - 1. Find a closed form for ana_n and evaluate a10a_{10}.

Example 17

challenge
A sequence is a1=1a_1 = 1 and an=an−11+an−1a_n = \frac{a_{n-1}}{1 + a_{n-1}}. Find ana_n in closed form.

Example 18

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

Example 19

easy
Identify whether an=5+3(n−1)a_n = 5 + 3(n-1) is recursive or explicit.

Example 20

easy
Write an explicit formula for the arithmetic sequence −3,1,5,9,…-3, 1, 5, 9, \ldots
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