Arithmetic and Geometric Sequences

A sequence is a function on the positions

$$3, \; 7, \; 11, \; 15, \; 19, \; \dots$$

Number the terms: the 1st is $3$, the 2nd is $7$, the 3rd is $11$. That pairing of a position with a value is a function — the input is $n$, the position, and the output is $a_n$, the term.

The only unusual thing is the domain. You can ask for term $4$ but not term $4.7$, so the inputs are whole numbers rather than every real number. That is what makes it a sequence rather than an ordinary function.

The two families

Type	Each term comes from	Fixed number
arithmetic	adding	the common difference $d$
geometric	multiplying	the common ratio $r$

Test a sequence by subtracting consecutive terms, then by dividing them — exactly the test that separates linear from exponential.

$$3, 7, 11, 15 \;\Rightarrow\; \text{differences } 4, 4, 4 \;\Rightarrow\; \textbf{arithmetic}, \; d = 4$$ $$5, 10, 20, 40 \;\Rightarrow\; \text{ratios } 2, 2, 2 \;\Rightarrow\; \textbf{geometric}, \; r = 2$$

A sequence can be neither. $1, 4, 9, 16$ has differences $3, 5, 7$ and ratios $4, 2.25, 1.78$ — no constant either way.

Recursive form: from the term before

A recursive rule says how to get the next term, plus where to start.

$$a_1 = 3, \qquad a_n = a_{n-1} + 4$$

Both lines are needed. The rule alone describes every sequence that goes up by $4$; the starting value picks out which one.

Type	Recursive rule
arithmetic	$a_n = a_{n-1} + d$
geometric	$a_n = a_{n-1} \cdot r$

Recursive form matches how a situation is usually described — each year the population grows by 3% — which makes it the natural first way to write a model down.

Its weakness is term $100$. Getting there means computing the $99$ terms before it.

Explicit form: straight from the position

An explicit rule computes any term directly.

$$a_n = a_1 + (n - 1)d \qquad a_n = a_1 \cdot r^{\,n-1}$$

The $n - 1$ is the part worth understanding rather than memorizing. Term $1$ has taken no steps yet. Reaching term $n$ takes $n - 1$ steps from the start, so that is how many times $d$ is added or $r$ is multiplied.

Term	Steps taken	Value
$a_1$	$0$	$3$
$a_2$	$1$	$3 + 4$
$a_3$	$2$	$3 + 8$
$a_n$	$n-1$	$3 + 4(n-1)$

Now term $100$ is one line:

$$a_{100} = 3 + 4(99) = 399$$

Why these are the functions you already know

Expand the arithmetic formula:

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

That is $y = mx + b$ with $d$ as the slope. An arithmetic sequence is a linear function sampled at the whole numbers, which is why its points lie on a straight line.

The geometric formula rearranges the same way:

$$a_n = a_1 \cdot r^{\,n-1} = \frac{a_1}{r} \cdot r^{\,n}$$

That is $y = a \cdot b^x$ — an exponential function, sampled at the whole numbers.

Sequence	Function	Graph
arithmetic	linear	points on a line
geometric	exponential	points on a curve

So everything already established about slopes and growth factors transfers without a new argument. The common difference is a slope; the common ratio is a growth factor.

Translating between the forms

Recursive to explicit: read off $a_1$ and the fixed number, then substitute.

$$a_1 = 6, \; a_n = a_{n-1} - 2 \quad\Rightarrow\quad a_n = 6 - 2(n-1)$$

Explicit to recursive: read off $a_1$ by putting $n = 1$, and take the fixed number from the formula.

$$a_n = 5 \cdot 3^{\,n-1} \quad\Rightarrow\quad a_1 = 5, \; a_n = 3a_{n-1}$$

Worked examples

Common mistakes

Practice problems