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 $n$ th 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.

Showing a random 20 of 50 problems.

Example 1

medium

Convert

a_1 = 3

a_n = 2a_{n-1}

into an explicit formula (start

n=1

Example 2

easy

For the explicit formula

a_n = (-1)^{n}

, list the first five terms.

Example 3

medium

Given

a_1 = 0

and

a_n = a_{n-1} + 2n

, find

a_5

Example 4

medium

A sequence is defined by

a_1 = 2

and

a_n = 2a_{n-1} + 1

. Find a closed-form explicit formula.

Example 5

hard

A sequence is defined by

a_1 = 1

and

a_n = a_{n-1} + n

. Find an explicit formula.

Example 6

easy

A sequence is given by

a_1 = 5

and

a_n = a_{n-1} + 3

. Find

a_2

Example 7

easy

A sequence is

a_1 = 1

a_n = a_{n-1} + n

. Find

a_4

Example 8

easy

Write the explicit formula for the arithmetic sequence

3, 7, 11, 15, \ldots

(start at

n=1

Example 9

hard

Given

a_1 = 100

and

a_n = 0.9 a_{n-1} + 10

, find

\lim_{n \to \infty} a_n

Example 10

medium

A sequence satisfies

a_1 = 2

a_n = 3a_{n-1}

. Find the explicit formula and identify the growth type.

Example 11

challenge

For the recursion

a_n = 2 a_{n-1} - a_{n-2}

with

a_1 = 3

a_2 = 5

, find

a_n

in closed form.

Example 12

medium

A sequence is defined by

a_n = n^2 - n

. For which

n

does

a_n = 12

Example 13

medium

For

a_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

a_n = 3 a_{n-1}

with

a_1 = 5

for an explicit formula.

Example 16

challenge

A sequence satisfies

a_1 = 1

a_n = a_{n-1} + 2n - 1

. Find a closed form for

a_n

and evaluate

a_{10}

Example 17

challenge

A sequence is

a_1 = 1

and

a_n = \frac{a_{n-1}}{1 + a_{n-1}}

. Find

a_n

in closed form.

Example 18

easy

Write a recursive formula for

5, 10, 20, 40, 80, \ldots

Example 19

easy

Identify whether

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, \ldots

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Related Concepts

Sequence Arithmetic Sequence Geometric Sequence Sigma Notation Series