Sequences as Functions | Moosa Academy

Sequences as Functions — Moosa Academy

A sequence looks like a list, but it behaves like a function. Feed it a position number and it returns the term sitting there. Seeing it that way explains why an arithmetic sequence graphs as points in a straight line — and lets you borrow the tools you already have for linear functions.

Concept A sequence is a function

A sequence is a function whose domain is the natural numbers (or part of them) and whose range is a set of real numbers. The domain supplies the position

1, 2, 3, \ldots, n

; the range supplies the terms

a_1, a_2, a_3, \ldots, a_n

Because the domain is whole numbers only, the graph is a set of separate dots — never a continuous line.

Finite sequence —

3,\ 6,\ 9,\ 12,\ 15

a fixed number of terms

Infinite sequence —

3,\ 6,\ 9,\ 12,\ 15,\ \ldots

continues without end

Theorem Arithmetic sequences

A sequence is arithmetic when each term is found by adding the same fixed number to the term before it. That number is the common difference $d$ .

d = +3

gives

3,\ 6,\ 9,\ 12,\ 15

— the terms climb

d = -4

gives

18,\ 14,\ 10,\ 6,\ 2,\ \ldots

— the terms fall

Plotted against position, the dots line up perfectly straight. The common difference plays exactly the role slope plays for a line: it is the constant step from one point to the next.

Example Is it arithmetic?

Test each sequence by subtracting consecutive terms.

a) $5,\ -6,\ -17,\ -28,\ \ldots$

-6 - 5 = -11

-17 - (-6) = -11

-28 - (-17) = -11

The difference is constant at $-11$ — arithmetic

b) $-4,\ 12,\ 28,\ 42,\ \ldots$

12 - (-4) = 16

28 - 12 = 16

42 - 28 = 14

The difference changes — not arithmetic

One mismatched difference is enough. Every gap must be identical, not merely the first few.

Example Extending a sequence

A sequence begins $18,\ 14,\ 10$ . Find the next four terms.

Common difference:

14 - 18 = -4

10 + (-4) = 6

6 + (-4) = 2

2 + (-4) = -2

-2 + (-4) = -6

$18,\ 14,\ 10,\ 6,\ 2,\ -2,\ -6$ with $d = -4$

Plotted against position these seven dots fall in a straight descending line, one step of $4$ down for each step of $1$ across.

Example Reaching a distant term

At a scout gathering the first row seats $3$ people, and every row after that seats $2$ more than the one before. How many people are in row $14$ ?

Treat position as

x

and the count as

y

: the point

(1,\ 3)

with slope

2

Point-slope form:

y - 3 = 2(x - 1)

Simplify:

y = 2x - 2 + 3

, so

y = 2x + 1

Substitute

x = 14

y = 2(14) + 1 = 29

Row 14 seats $29$ people

Counting up thirteen times would work but invites mistakes. The formula reaches any position in one step, which is the real payoff of treating the sequence as a function.

Note Dots, not a line

It is tempting to join the points up, but position $2.5$ has no meaning — there is no row two-and-a-half. The straight line is a guide the eye supplies; the sequence itself is only the dots sitting on it.

Summary

A sequence is a function from position numbers to terms.
The domain is $1, 2, 3, \ldots$ ; the range is $a_1, a_2, a_3, \ldots$
A finite sequence stops; an infinite one continues without end.
A sequence is arithmetic when every consecutive difference is the same value $d$ .
Test by subtracting consecutive terms — a single mismatch rules it out.
Arithmetic sequences graph as separate dots lying on a straight line, with $d$ acting as the slope.
Point-slope form reaches a distant term in one step instead of counting up.