Sequences

Pre-Calculus · Axiom Academy

A sequence is just a function on the counting numbers — feed it a position, it hands you a term. Feed in a position — 1st, 2nd, 3rd — and the sequence returns the number in that slot. That is exactly what a function does: it maps each input to one output. The only restriction is the domain. 2. Explicit Formulas — Jump Straight to a Term An explicit formula writes the n th term directly in terms of n . Hand it any position and it returns that term in one step — no need to know any other term first. The sequence has the explicit rule a_n = 2n - 1 . Plug in a position and read off the term: 3. Recursive Formulas — Build on the Term Before A recursive formula defines each term using the previous one (or two). It reads like an instruction: "to get the next term, do this to the current one." It needs two pieces — a starting term and a step rule. then add the same step each time Start at a_1 = 5 and repeatedly add 3. The chain comes out one link at a time: Reach any term in one step ( a_ 100 instantly), but the pattern of growth is less obvious. Shows the growth rule plainly, but reaching a_ 100 means computing all 99 terms before it. Some sequences show up so often they earn names. Notice each is captured by an explicit or recursive rule — sometimes both. Each term is the sum of the two before it — a recursive rule, seen throughout nature. Dots that stack into a triangle: term n equals . The squares of the counting numbers — the backbone of quadratic patterns.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.