Recursive vs Explicit Formulas Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Recursive vs Explicit Formulas.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A recursive formula builds each term from the one before; an explicit formula jumps straight to the nth term from n.

Common stuck point: The procedure for recursive vs explicit formulas is the easy part; the trap is giving a recursive rule with no initial condition. Asking "Does the rule compute a term from the term(s) before it (recursive), or straight from the position $n$ (explicit)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the rule compute a term from the term(s) before it (recursive), or straight from the position $n$ (explicit)?

Worked Examples

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}

Answer

a_n = 5n-2

;

a_{50} = 248

First step

First few terms:

3, 8, 13, 18, \ldots

— arithmetic with

d=5

Full solution

2
Explicit: $a_n = 3 + (n-1) \cdot 5 = 5n - 2$ .
3
Verify: $a_1 = 3$ ✓, $a_2 = 8$ ✓.
4
$a_{50} = 5(50)-2 = 248$ .

A constant-difference recursion defines an arithmetic sequence. The explicit formula

a_n = a_1 + (n-1)d

lets you reach any term instantly.

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

medium

A sequence is defined by

a_1 = 5

a_n = a_{n-1} + 3

. Write the explicit formula and find

a_{20}

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

medium

Given

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

with

a_1 = 1

, find an explicit formula for

a_n

Example 6

hard

A sequence is defined by

a_1 = 1

and

a_n = a_{n-1} + n

. Find an explicit formula.

Example 7

hard

Find an explicit formula for

a_n

given

a_1 = 1

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

Example 8

challenge

Find a closed-form formula for the Fibonacci numbers

F_n

with

F_1 = F_2 = 1

Example 9

challenge

The Tower of Hanoi puzzle satisfies

T_1 = 1

and

T_n = 2 T_{n-1} + 1

. Find an explicit formula for

T_n

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Write a recursive formula for

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

Example 2

medium

Convert

a_n = 4n - 1

to a recursive formula.

Example 3

easy

A sequence is given by

a_1 = 5

and

a_n = a_{n-1} + 3

. Find

a_2

Example 4

easy

Given the explicit formula

a_n = 4n - 1

, find

a_3

Example 5

easy

A sequence has

a_1 = 2

and

a_n = 2a_{n-1}

. Find

a_3

Example 6

easy

Write the explicit formula for the arithmetic sequence

3, 7, 11, 15, \ldots

(start at

n=1

Example 7

easy

A sequence is

a_1 = 1

a_n = a_{n-1} + n

. Find

a_4

Example 8

easy

Given

a_n = 2^n

, find

a_5

Example 9

easy

Identify whether

a_n = 5 + 3(n-1)

is recursive or explicit.

Example 10

easy

A sequence has

a_0 = 1

and

a_n = a_{n-1} + 2

. Find

a_2

Example 11

medium

Convert the recursive sequence

a_1 = 4

a_n = a_{n-1} + 6

into an explicit formula (start

n=1

Example 12

medium

Convert

a_1 = 3

a_n = 2a_{n-1}

into an explicit formula (start

n=1

Example 13

medium

A sequence is

a_1 = 7

a_n = a_{n-1} + 5

. Use the explicit form to find

a_{20}

Example 14

medium

The Fibonacci sequence has

F_1 = 1

F_2 = 1

F_n = F_{n-1} + F_{n-2}

. Find

F_6

Example 15

medium

A sequence is defined by

a_n = n^2 - n

. For which

n

does

a_n = 12

Example 16

medium

A sequence satisfies

a_1 = 100

and

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

. Find

a_4

Example 17

medium

Convert

a_n = 6n - 2

back to a recursive formula (start

n=1

Example 18

medium

A sequence has

a_1 = 2

a_n = 3a_{n-1} + 1

. Find

a_3

Example 19

medium

For

a_n = 3n + 1

, find the sum of the first 4 terms.

Example 20

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 21

challenge

Find a closed form for

a_1 = 2

a_n = 3a_{n-1} + 1

Example 22

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 23

easy

Given

a_1 = 7

and

a_n = a_{n-1} - 2

, find

a_5

Example 24

easy

Write an explicit formula for the arithmetic sequence

-3, 1, 5, 9, \ldots

Example 25

easy

Write a recursive formula for the sequence

a_n = 3n + 2

Example 26

easy

For the explicit formula

a_n = (-1)^{n}

, list the first five terms.

Example 27

easy

For

a_1 = 4

and

a_n = -a_{n-1}

, find

a_6

Example 28

medium

A sequence has

a_1 = 3

and

a_n = 2a_{n-1} + 1

. Find

a_4

Example 29

medium

Convert

a_n = 5 \cdot 2^{n-1}

to a recursive formula.

Example 30

medium

Given

a_1 = 0

and

a_n = a_{n-1} + 2n

, find

a_5

Example 31

medium

For the sequence

1, 4, 9, 16, 25, \ldots

, write an explicit formula.

Example 32

hard

Solve the recurrence

a_n = 3 a_{n-1}

with

a_1 = 5

for an explicit formula.

Example 33

hard

Given

a_1 = 100

and

a_n = 0.9 a_{n-1} + 10

, find

\lim_{n \to \infty} a_n

Example 34

hard

A sequence satisfies

a_n = a_{n-1} + 3

and

a_5 = 18

. Find

a_1

and an explicit formula.

Example 35

hard

A bank account starts at $1000 and each year multiplies by

1.05

then adds $200. Write a recursive formula and compute the balance after 3 years.

Example 36

hard

A sequence is defined by

a_1 = 4

a_n = a_{n-1}/2

. After how many terms does

a_n

first drop below

0.01

Example 37

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.

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Related Concepts

Sequence Arithmetic Sequence Geometric Sequence Sigma Notation Series

Background Knowledge

These ideas may be useful before you work through the harder examples.

sequencearithmetic sequencegeometric sequence