Introduction to Arithmetic Progression (AP

What is a Sequence? A sequence is an ordered list of numbers arranged according to a definite rule. The numbers in a sequence are called its terms, and we name them $t_1, t_2, t_3, \dots$ (or $a_1, a_2, a_3, \dots$) according to their position. For example, in $2, 4, 6, 8, 10, \dots$ each term is $2$ more than the one before it.

What is a Series? A series is what you get when the terms of a sequence are added: $2 + 4 + 6 + 8 + 10 + \dots$ A sequence simply lists the numbers; a series totals them. This chapter is built around one very common and useful kind of sequence.

What is an Arithmetic Progression (AP)? An Arithmetic Progression is a sequence in which each term, except the first, is obtained by adding a fixed number to the previous term. That fixed number is called the common difference, written $d$. Because the same amount is added every time, the terms move in equal steps.

The defining condition. For a list $t_1, t_2, t_3, t_4, \dots$ to be an AP, the difference between each term and the one before it must be the same: $t_2 - t_1 = t_3 - t_2 = t_4 - t_3 = \dots = d$. Notice the subtraction is always later term minus earlier term ($t_{k+1} - t_k$), never the other way round. Getting this order wrong is the single most common mistake, because it flips the sign of $d$.

How to test a list for AP. Take consecutive differences across the whole list. If every difference comes out equal, it is an AP and that common value is $d$. If even one difference disagrees, the list is not an AP. Checking just the first two terms is not enough — a list like $2, 4, 6, 10$ passes the first check but fails at $10 - 6 = 4 \ne 2$, sorry, $6-4=2$ but $4-2=2$ and $10-6=4$, so it is not an AP.

The three flavours of AP. If $d > 0$ the terms grow — an increasing AP (e.g. $3, 7, 11, 15$). If $d < 0$ the terms shrink — a decreasing AP (e.g. $20, 17, 14, 11$). If $d = 0$ every term is identical (e.g. $5, 5, 5, 5$); this is a constant AP, and yes it still counts as an AP.

Finite and infinite APs. An AP with a definite number of terms ($3, 6, 9, 12, 15$) is finite and has a last term. An AP that continues without end ($3, 6, 9, \dots$) is infinite and has no last term. The three dots "$\dots$" signal that the pattern carries on.

Why APs matter. Equal-step patterns appear everywhere: a salary that rises by a fixed raise each year, the seats in successive rows of a hall, the rungs of a ladder, the money saved each month under a fixed-increment plan, even the cost of a taxi where each kilometre adds a flat charge. Recognising an AP lets you jump straight to any term or total without listing everything.

• A monthly salary that increases by a fixed amount every year.
• Seats per row in an auditorium that grow by a constant from front to back.
• The water level in a tank falling by a fixed amount each hour.
• The amount repaid each year on a loan settled in equal-step instalments.

Common mistakes to avoid. (1) Subtracting in the wrong direction and reporting $+d$ as $-d$. (2) Declaring a list an AP after checking only one gap. (3) Assuming a "nice looking" list such as $1, 4, 9, 16$ (squares) or $2, 4, 8, 16$ (doubling) is an AP — neither is, because their differences are not constant. Always verify with actual subtraction.