What is the difference between a recursive and an explicit formula?

A recursive formula tells you where to start and how to get from one term to the next, so you must know the previous term to use it. An explicit formula computes any term directly from its position number. Recursive is faster for listing the next two or three terms; explicit is the only practical way to find something like the 200th term.

Why is there an n-1 in the explicit formulas instead of n?

Because you start counting from a_1, not from a step zero. Getting to the 5th term from the 1st takes only 4 additions (or 4 multiplications), so getting to the nth term takes n-1 of them. Quick check: substitute n = 1 into your formula. If it does not return the first term, your exponent or coefficient is shifted.

Can a sequence be both arithmetic and geometric?

Yes, but only in the constant case. A sequence like 7, 7, 7, 7, has d = 0 and r = 1, so it satisfies both definitions. Apart from constant sequences (and ignoring sequences of all zeros, where the ratio is undefined), no sequence is both.

How do I know a sequence is neither arithmetic nor geometric?

Compute all the consecutive differences and all the consecutive ratios. If neither list is constant, the sequence is neither type. Squares like 1, 4, 9, 16 and the Fibonacci sequence 1, 1, 2, 3, 5, 8 are common examples — Fibonacci does have a recursive rule, a_n = a_n-1 + a_n-2, but it is not arithmetic because the amount added changes each time.

ALG1-10.3

Arithmetic & Geometric Sequences

Learn to tell arithmetic from geometric sequences, write recursive and explicit formulas, and see how they connect to linear and exponential functions.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Arithmetic & Geometric Sequences, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Arithmetic & Geometric Sequences free

What this lesson covers

Every sequence is just a list of numbers in order, but the interesting question is always the same: what did you do to one term to get the next? If you added the same amount every time, the sequence is arithmetic. If you multiplied by the same amount every time, it is geometric. That one decision drives everything else in this lesson.

Once you know the type, you can describe the sequence two ways: recursively, by saying where it starts and how to step forward, or explicitly, by writing a formula that jumps straight to the 50th term without listing the first 49. Along the way you will see something that ties this unit back to earlier work — arithmetic sequences are linear functions in disguise, and geometric sequences are exponential functions in disguise. Same math, new notation.

Reading a Sequence: Differences and Ratios

A sequence is an ordered list of numbers. Each number is a term, and terms are labeled by position:

a_1

is the first term,

a_2

the second, and

a_n

the term in position

n

. The position number

n

is always a counting number — there is no term number 0.5 and (in this course) no

a_0

unless a problem sets one up on purpose.

To classify a sequence, run two quick tests on consecutive terms. Subtract to look for a constant common difference

d

, then divide to look for a constant common ratio

r

. Whichever test gives the same answer every time tells you the type.

Sequence	Differences	Ratios	Type
$4, 9, 14, 19$	$+5, +5, +5$	$2.25, 1.56, \ldots$	Arithmetic, $d = 5$
$3, 6, 12, 24$	$+3, +6, +12$	$2, 2, 2$	Geometric, $r = 2$
$80, 60, 40, 20$	$-20, -20, -20$	$0.75, 0.67, \ldots$	Arithmetic, $d = -20$
$1, 4, 9, 16$	$+3, +5, +7$	$4, 2.25, 1.78$	Neither

Two cautions. First, always subtract in the direction later term minus earlier term, so

d = a_2 - a_1

; reversing the order flips the sign and sends the rest of the problem the wrong way. Second, check more than one pair. The list

2, 4, 8, 10

has a ratio of 2 for the first two steps and then quits, so it is not geometric. A sequence is only arithmetic or geometric if the pattern holds for every consecutive pair.

Recursive Formulas: Start Here, Then Do This

A recursive formula describes a sequence in terms of the term before it. It always has two parts, and leaving one out makes the formula useless.

For an arithmetic sequence:

a_1 = \text{first term}

and

a_n = a_{n-1} + d

for

n \ge 2

.

For a geometric sequence:

a_1 = \text{first term}

and

a_n = a_{n-1} \cdot r

for

n \ge 2

.

So the sequence

6, 10, 14, 18, \ldots

is written

a_1 = 6

a_n = a_{n-1} + 4

. The sequence

5, 15, 45, \ldots

is written

a_1 = 5

a_n = a_{n-1} \cdot 3

.

The notation

a_{n-1}

trips people up. It does not mean "the term

a_n

minus 1." The subscript is an address, not a quantity:

a_{n-1}

is the term one position earlier. If

n = 7

, then

a_{n-1}

means

a_6

.

The most common mistake is writing only

a_n = a_{n-1} + 4

and stopping. That rule fits

6, 10, 14, \ldots

but it also fits

1, 5, 9, \ldots

and infinitely many other lists. Without the starting value the sequence is not pinned down. The second most common mistake is mixing operations — writing

a_n = a_{n-1} + 3

for a sequence that actually triples. Say the pattern out loud ("times three") before you write it.

Recursive formulas are excellent for generating the next few terms and terrible for finding

a_{60}

, because you would have to compute all 59 terms before it. That is exactly the problem explicit formulas solve.

Explicit Formulas: Jump Straight to Any Term

An explicit formula gives

a_n

directly from

n

. Build it by counting how many steps you take from the first term.

To reach

a_5

from

a_1

you take 4 steps. To reach

a_n

you take

n - 1

steps. So:

a_n = a_1 + (n-1)d \qquad \text{arithmetic}

a_n = a_1 \cdot r^{\,n-1} \qquad \text{geometric}

For

6, 10, 14, 18, \ldots

with

a_1 = 6

and

d = 4

a_n = 6 + (n-1)4

, which simplifies to

a_n = 4n + 2

. Check it —

a_3 = 4(3) + 2 = 14

, correct.

For

5, 15, 45, \ldots

with

a_1 = 5

and

r = 3

a_n = 5 \cdot 3^{n-1}

. Then

a_6 = 5 \cdot 3^5 = 5 \cdot 243 = 1215

.

The number one error in this whole lesson is the off-by-one: writing

a_n = 6 + 4n

a_n = 5 \cdot 3^n

. Both formulas are shifted one term too far. There is a five-second fix — plug in

n = 1

. A correct formula must return the first term.

5 \cdot 3^1 = 15

, not 5, so that version is wrong.

In the geometric formula, the exponent applies only to

r

, never to the product.

5 \cdot 3^{n-1}

means

5

times

3^{n-1}

; it does not mean

(5 \cdot 3)^{n-1} = 15^{n-1}

. Order of operations handles the exponent first, then the multiplication.

Sequences Are Functions: The Linear and Exponential Link

A sequence is a function whose inputs are the position numbers

1, 2, 3, \ldots

and whose outputs are the terms. That reframing explains why these formulas look so familiar.

Distribute the arithmetic formula:

a_n = a_1 + (n-1)d = dn + (a_1 - d)

. That is

y = mx + b

with slope

d

. The common difference is the slope — the constant rate of change per step. Rewrite the geometric formula as

a_n = \left(\frac{a_1}{r}\right) r^{n}

and you get

y = ab^x

, an exponential function with base

r

. The common ratio is the growth factor.

Feature	Arithmetic	Geometric
Step rule	add $d$	multiply by $r$
Explicit form	$a_n = a_1 + (n-1)d$	$a_n = a_1 r^{n-1}$
Function family	linear	exponential
Graph shape	points along a line	points along a curve
Grows by	equal amounts	equal percents

One genuine difference: the graph of a sequence is a set of separate dots, not a connected line or curve, because the domain is only the counting numbers. There is no term between

a_3

and

a_4

, so you never connect the points.

This link is practical. A phone plan that charges a flat 15 dollars more each month is arithmetic and linear. A bacteria culture that doubles every hour is geometric and exponential. When someone tells you a quantity grows by 5 percent per year, the multiplier is

r = 1.05

, so successive years form a geometric sequence — and that is the same structure you will use for compound interest and half-life problems later.

Working Backward: Finding $d$ , $r$ , or the Term Number

Problems often hand you two terms that are not next to each other, or ask which position a given value occupies.

Suppose an arithmetic sequence has

a_3 = 11

and

a_7 = 27

. Going from position 3 to position 7 is 4 steps, and the total change is

27 - 11 = 16

, so

d = 16 \div 4 = 4

. Back up two steps from

a_3

to get

a_1 = 11 - 2(4) = 3

. The explicit formula is

a_n = 3 + (n-1)4 = 4n - 1

.

For geometric sequences, the same idea uses division. If

a_2 = 12

and

a_5 = 96

, that is 3 steps, so

r^3 = 96 \div 12 = 8

and

r = 2

. Then

a_1 = 12 \div 2 = 6

and

a_n = 6 \cdot 2^{n-1}

.

To find a term's position, set the explicit formula equal to the value and solve. Is 71 in the sequence

4n - 1

? Solve

4n - 1 = 71

to get

n = 18

, a counting number, so yes — it is the 18th term. Is 50 in that sequence?

4n - 1 = 50

gives

n = 12.75

, which is not a valid position, so 50 never appears. That whole-number check is the point of the question.

Watch for decreasing geometric sequences:

64, 32, 16, \ldots

has

r = \frac{1}{2}

, a ratio between 0 and 1, and it decreases without ever reaching zero. A negative ratio makes signs alternate —

3, -6, 12, -24

has

r = -2

Key terms

Sequence.: An ordered list of numbers, treated as a function whose inputs are the position numbers $1, 2, 3, \ldots$ and whose outputs are the terms.
Term ( $a_n$ ).: A single number in a sequence, labeled by its position. $a_4$ means the fourth term; the subscript is an address, not a multiplier.
Arithmetic sequence.: A sequence in which the same number is added to each term to get the next, giving a constant common difference.
Common difference ( $d$ ).: The constant value $a_n - a_{n-1}$ in an arithmetic sequence; it equals the slope of the matching linear function.
Geometric sequence.: A sequence in which each term is multiplied by the same nonzero number to get the next, giving a constant common ratio.
Common ratio ( $r$ ).: The constant value $a_n \div a_{n-1}$ in a geometric sequence; it equals the base (growth factor) of the matching exponential function.
Recursive formula.: A definition that states the first term and a rule for producing each term from the one before, such as $a_1 = 6$ and $a_n = a_{n-1} + 4$ .
Explicit formula.: A formula that computes $a_n$ directly from the position $n$ , such as $a_n = a_1 + (n-1)d$ or $a_n = a_1 r^{n-1}$ .

Worked example

The first four terms of a sequence are 96, 48, 24, 12. Classify the sequence, write both a recursive and an explicit formula, find the 8th term, and determine whether 1.5 is a term of the sequence.

Step 1 — Classify. Check differences:

48 - 96 = -48

but

24 - 48 = -24

. Not constant, so it is not arithmetic. Check ratios:

48 \div 96 = \frac{1}{2}

24 \div 48 = \frac{1}{2}

12 \div 24 = \frac{1}{2}

. The ratio is constant, so the sequence is geometric with

r = \frac{1}{2}

and

a_1 = 96

.

Step 2 — Recursive formula. State both parts:

a_1 = 96

and

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

for

n \ge 2

. (Writing

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

is equally correct.)

Step 3 — Explicit formula. Use

a_n = a_1 r^{n-1}

, so

a_n = 96\left(\frac{1}{2}\right)^{n-1}

. Verify with

n = 1

96\left(\frac{1}{2}\right)^0 = 96 \cdot 1 = 96

. That matches the first term, so the exponent is right.

Step 4 — Find $a_8$ .

a_8 = 96\left(\frac{1}{2}\right)^{7} = 96 \cdot \frac{1}{128} = \frac{96}{128} = 0.75

.

Step 5 — Is 1.5 a term? Set

96\left(\frac{1}{2}\right)^{n-1} = 1.5

. Divide both sides by 96:

\left(\frac{1}{2}\right)^{n-1} = \frac{1.5}{96} = \frac{1}{64}

. Since

\left(\frac{1}{2}\right)^6 = \frac{1}{64}

, we need

n - 1 = 6

, so

n = 7

. Because 7 is a counting number, yes — 1.5 is the 7th term. (Listing forward confirms it: 96, 48, 24, 12, 6, 3, 1.5.)

Practice questions

Which explicit formula generates the sequence

7, 11, 15, 19, \ldots

$a_n = 7 + 4n$
$a_n = 4n + 3$
$a_n = 7 \cdot 4^{n-1}$
$a_n = 4 + 7(n-1)$

Answer: $a_n = 4n + 3$

The differences are all

+4

, so the sequence is arithmetic with

d = 4

and

a_1 = 7

. Then

a_n = 7 + (n-1)4 = 7 + 4n - 4 = 4n + 3

. Test it:

a_1 = 4(1) + 3 = 7

and

a_3 = 4(3)+3 = 15

, both correct. The choice

7 + 4n

is the classic off-by-one — it gives 11 for

n = 1

, one term too far along. The third choice multiplies instead of adds, and the fourth swaps the roles of the first term and the common difference.

A geometric sequence has

a_2 = 18

and

a_4 = 162

. Find the common ratio and

a_1

, then write the explicit formula. Assume

r

is positive.

Answer: $r = 3$ , $a_1 = 6$ , and $a_n = 6 \cdot 3^{n-1}$ .

Going from position 2 to position 4 is two multiplications by

r

, so

18 \cdot r^2 = 162

, giving

r^2 = 9

and

r = 3

(the problem says

r

is positive, which rules out

-3

). Back up one step from

a_2

a_1 = 18 \div 3 = 6

. Substituting into

a_n = a_1 r^{n-1}

gives

a_n = 6 \cdot 3^{n-1}

. Check

a_4 = 6 \cdot 27 = 162

. Students often divide

162

18

and call the result the ratio, but

9

is the two-step multiplier, not the one-step ratio.

Explain why the sequence

5, 10, 15, 20, \ldots

corresponds to a linear function while

5, 10, 20, 40, \ldots

corresponds to an exponential function, and describe how their graphs differ.

Answer: The first is arithmetic with $d = 5$ , so $a_n = 5n$ , a linear rule with constant rate of change 5. The second is geometric with $r = 2$ , so $a_n = 5 \cdot 2^{n-1}$ , an exponential rule with a constant growth factor. Both graphs are sets of discrete points, but the first set lies along a straight line and the second lies along an increasing curve that gets steeper.

In the first sequence you add the same amount at every step, which is exactly what constant slope means in a linear function. In the second you multiply by the same amount at every step, which is what a constant base means in an exponential function. The key detail people forget is the domain: the inputs are only

n = 1, 2, 3, \ldots

, so you plot separate dots and never connect them — there is no term between the 2nd and the 3rd.

FAQ

What is the difference between a recursive and an explicit formula?: A recursive formula tells you where to start and how to get from one term to the next, so you must know the previous term to use it. An explicit formula computes any term directly from its position number. Recursive is faster for listing the next two or three terms; explicit is the only practical way to find something like the 200th term.
Why is there an $n-1$ in the explicit formulas instead of $n$ ?: Because you start counting from $a_1$ , not from a step zero. Getting to the 5th term from the 1st takes only 4 additions (or 4 multiplications), so getting to the $n$ th term takes $n-1$ of them. Quick check: substitute $n = 1$ into your formula. If it does not return the first term, your exponent or coefficient is shifted.
Can a sequence be both arithmetic and geometric?: Yes, but only in the constant case. A sequence like $7, 7, 7, 7, \ldots$ has $d = 0$ and $r = 1$ , so it satisfies both definitions. Apart from constant sequences (and ignoring sequences of all zeros, where the ratio is undefined), no sequence is both.
How do I know a sequence is neither arithmetic nor geometric?: Compute all the consecutive differences and all the consecutive ratios. If neither list is constant, the sequence is neither type. Squares like $1, 4, 9, 16$ and the Fibonacci sequence $1, 1, 2, 3, 5, 8$ are common examples — Fibonacci does have a recursive rule, $a_n = a_{n-1} + a_{n-2}$ , but it is not arithmetic because the amount added changes each time.

Learn this with a teacher, not a page

The Crimsora tutor teaches Arithmetic & Geometric Sequences live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.

Try a live session free

What you'll do in this lesson

What this lesson covers

Reading a Sequence: Differences and Ratios

Recursive Formulas: Start Here, Then Do This

Explicit Formulas: Jump Straight to Any Term

Sequences Are Functions: The Linear and Exponential Link

Working Backward: Finding ddd, rrr, or the Term Number

Key terms

Worked example

Practice questions

FAQ

Learn this with a teacher, not a page

Working Backward: Finding $d$ , $r$ , or the Term Number