AI Tutor
If you save $5 every day versus doubling a penny every day, both create an ordered list of numbers called a sequence. But when you chart them side by side, their visual footprints look completely different.
A sequence is really just a function where the input nn is the term position (1st, 2nd, 3rd...), and the output ana_n is the actual value of that term.
Because term positions only count in whole steps (n=1,2,3,…n = 1, 2, 3, \dots), how does that change the way we plot them on a coordinate plane?

Sequences as Discrete Functions

Standard algebra functions use a continuous domain of all real numbers, but a sequence has a discrete domain restricted to natural numbers (n∈{1,2,3,… }n \in \{1, 2, 3, \dots\}).
This means you never connect the plotted points with a solid line or curve — there is no such thing as a "2.5th term" in a sequence.
Now that we know sequences look like isolated points, what specific shape do those points create when each term increases by adding the same amount?

Graphing Arithmetic Sequences

An arithmetic sequence adds a fixed constant called the common difference (dd) to get each new term, matching the explicit formula an=a1+(n−1)da_n = a_1 + (n - 1)d.
If you rewrite this as an=d⋅n+(a1−d)a_n = d \cdot n + (a_1 - d), it takes the exact form of linear slope-intercept form (y=mx+by = mx + b), meaning the points form a straight linear line with slope m=dm = d.