Sequences as Functions
A sequence looks like a list but behaves like a function: feed it a position and it returns a term. Seeing it that way explains why an arithmetic sequence graphs as dots on a straight line.
--
A sequence looks like a list but behaves like a function: feed it a position and it returns a term. Seeing it that way explains why an arithmetic sequence graphs as dots on a straight line.
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.
Because the domain is whole numbers only, the graph is a set of separate dots — never a continuous line.
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 .
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.
Test each sequence by subtracting consecutive terms.
a)
b)
One mismatched difference is enough. Every gap must be identical, not merely the first few.
A sequence begins . Find the next four terms.
Plotted against position these seven dots fall in a straight descending line, one step of down for each step of across.
At a scout gathering the first row seats people, and every row after that seats more than the one before. How many people are in row ?
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.
It is tempting to join the points up, but position 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.