Arithmetic progression (AP) — nth term, sum of n terms — derivations: Foundations

This page assumes you have seen nothing. Before you touch the parent note, every squiggle it uses is unpacked below, in the order they must be learned.

0 · What is a "sequence"?

Picture a row of stepping-stones across a river. You cross them left to right, in order. That row is a sequence.

Read the figure. Each blue stone holds one number of the list $2, 5, 8, 11, 14$ ; the yellow tag under each stone is its position (1st, 2nd, …); the white arrows show the fixed left-to-right reading order. That ordered row is exactly what "sequence" means.

Why the topic needs it. An AP is a special kind of sequence, so we must first agree on what an ordinary list of ordered numbers even is.

1 · Subscripts: $a_{1}, a_{2}, a_{3}, \dots$ and the general $a_{n}$

(Those three dots "…" just mean "keep listing in the same obvious pattern"; they get their own proper definition in §5.)

Now the star of the show: the letter $n$ .

Read the figure. The four yellow stones are actual terms with their address labels $a_{1}, a_{2}, a_{3}, a_{4}$ in blue above them. The pink stone on the right holds a "?" — it is $a_{n}$ , the general term: hand it any position $n$ and it becomes one specific stone.

Why the topic needs it. The whole point of a formula is to describe every term at once. We cannot write out an infinite list, so we say "give me any position $n$ , and I'll hand you $a_{n}$ ." Without a symbol for "any position", no formula is possible.

2 · The building blocks $a$ and $d$

To find $d$ , you subtract a term from the one after it. Every gap gives the same value: $d = a_{2} - a_{1} and d = a_{3} - a_{2} and so on for every gap.$

Read the figure. The blue bars are the terms of $2, 5, 8, 11, 14$ drawn as heights; the pink arrow at the start marks $a = 2$ ; the double-headed pink arrow between two bars measures one step $d = 3$ . Because every bar is exactly $d$ taller than the one before, the staircase is perfectly even.

Why the topic needs it. The parent note's ONE idea is: everything about an AP is rebuilt from just two numbers, $a$ and $d$ . These two symbols are those two numbers. Master them and the rest is bookkeeping.

3 · The minus sign as "distance / difference"

The tiny expression $a_{n + 1}$ means "the term one position after $a_{n}$ " — because $n + 1$ is one address higher than $n$ . So $a_{n + 1} - a_{n}$ = "(next term) minus (this term)" = "the size of one step".

Why the topic needs it. The formal definition of an AP is the single equation $a_{n + 1} - a_{n} = d$ . If the minus sign is fuzzy, the definition is fuzzy.

4 · Why " $(n - 1)$ " and not " $n$ " — and the nth-term formula

This is the trap the parent note warns about, so we build it slowly.

Count on your fingers:

Reach term...	Jumps taken	$d$ 's added	Value
$a_{1}$	$0$	$0 d$	$a$
$a_{2}$	$1$	$1 d$	$a + d$
$a_{3}$	$2$	$2 d$	$a + 2 d$
$a_{n}$	$n - 1$	$(n - 1) d$	$a + (n - 1) d$

Read the last row straight off the table: start at $a$ , add the step $d$ a total of $(n - 1)$ times. That is the nth-term formula.

Why the topic needs it. This single "off-by-one" is the most common error in the whole topic. Getting the gap between position number and number of jumps straight now saves every later formula.

5 · A "sum", the ellipsis, and the symbol $S_{n}$

First, the three dots.

Now the total.

Why the topic needs it. Half the topic is about adding an AP quickly. $S_{n}$ is the name of that total, and $\sum$ is the shorthand you will meet again and again.

6 · The picture behind "average of first and last"

Read the figure. The blue dots are the terms of an AP climbing in a straight line; the pink dashed line is their average value. The two yellow double-arrows show the key fact: the first term sits below that line by exactly the same gap that the last term sits above it. That balance is why "average = $\frac{first + last}{2}$ " works.

Why the topic needs it. The prettiest sum formula, $S_{n} = n \times \frac{a + a _{n}}{2}$ , reads "how many terms × their average". That only works because AP terms are evenly spaced — this figure is why.

7 · "Linear in $n$ " — a straight-line list

First, two symbols this section uses.

Now the definition.

The nth-term formula $a_{n} = a + (n - 1) d = d \cdot n + (a - d)$ is of this shape: slope $d$ , intercept $(a - d)$ . So plotting term-value against position gives evenly-climbing dots on a straight line — see Linear Functions. This is why " $a_{n}$ linear in $n$ " is a dead giveaway that a sequence is an AP.

Recall Why quadratic

$S_{n}$ also signals an AP Adding up a straight line gives an $n^{2}$ shape (a quadratic) with no constant term. That is why $S_{n} = 3 n^{2} + 2 n$ in the parent's Example 3 turned out to be an AP. What kind of expression in $n$ is $a_{n}$ for an AP? ::: Linear (straight line, slope $= d$ ).

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer aloud.

What does the subscript in

a_{4}

tell you?

The position (4th term), it is an address, not a multiplication.

What does the letter

n

stand for?

Any whole counting number — a placeholder for "some position".

What is

a

(or

a_{1}

The first term, where the sequence starts.

What is the common difference

d

, and how do you find it?

The fixed step added each time; find it by subtracting a term from the next,

d = a_{2} - a_{1}

Can

d

be zero or negative?

Yes —

d = 0

gives a constant sequence,

d < 0

gives a decreasing AP; the rule is only that

d

is the same at every gap.

Why subtract (not divide) to get

d

An AP adds a constant, so the gap is a subtraction; dividing belongs to a GP.

To reach the

n

th term, how many steps of

d

do you take?

n - 1

, because the first term costs zero jumps.

Write the nth-term formula.

a_{n} = a + (n - 1) d

What does

S_{n}

mean?

The total of the first

n

terms added together.

What does the ellipsis "

\dots

" mean?

"Continue the same pattern up to the last written term."

What does

\sum_{k = 1}^{n} a_{k}

mean, and what is

k

"Add all terms

a_{k}

as the counter

k

runs

1

n

k

is a temporary counting label. It equals

S_{n}

What does

n^{2}

mean, and what does

do?

n^{2} = n \times n

; the square root undoes squaring, e.g.

9 = 3

Why is the average of an AP just

\frac{first + last}{2}

The terms are evenly spaced, so first and last are symmetric about the middle.

What does "

a_{n}

is linear in

n

" look like on a graph?

A straight line of climbing dots (slope

= d