Sequences as Functions of the Term Number

A function takes an input and gives one output. A sequence does exactly that, except its only legal inputs are the whole numbers 1, 2, 3, 4, and so on, never fractions or negatives. So a(1), a(2), a(3) are just function values, and a(n) is the rule that produces the nth term from its position n. Thinking of a sequence this way turns an abstract list of numbers into something you can graph: plot the term number n on the horizontal axis and the term value a(n) on the vertical axis.

For an arithmetic sequence, each term increases by the same common difference d, which means the points (n, a(n)) all lie perfectly on a straight line with slope d. That line has equation a(n) = dn + c for some constant c, and a(1) tells you where the line sits. But the sequence is not the line itself: the line is unbroken, while the sequence only exists at the dots sitting on it, one for each counting number. For a geometric sequence, each term is multiplied by a common ratio r instead of added to, so the points no longer sit on a line. The vertical gaps between consecutive dots keep growing or shrinking, tracing a curve instead of a straight path.

This graphing view connects directly to explicit formulas and recursive rules. The slope of the arithmetic line is the common difference, and the starting point is the first term, so reading a graph straight off gives you the formula. Spotting whether plotted points look straight or curved is also the fastest way to tell an arithmetic sequence from a geometric one before doing any algebra.