Sequence Notation

Algebra 2 · Axiom Academy

How a_n names the n th term of a list — and the two ways to define a whole sequence. 1. Every Term Has an Address: a_n Line the numbers up and number the seats. The term in seat n is written a_n — the subscript is its position , not a multiplier. Counting starts at n = 1 , so a_1 is the first term, a_2 the second, and so on. The sequence — an ordered list Its terms, each named by position 2. Explicit Formulas: Jump Straight to a_n An explicit formula gives a_n directly as a function of its position: a_n = f(n) . Drop in any n and out comes that term — no need to know the ones before it. Substitute the position for n in a_n = 2n + 1 : The rule f can be any function — it need not be linear 3. Recursive Definitions: Build From the Previous Term A recursive definition builds the list from the inside out: give a starting value, then a rule for getting each term from the one before it. a_1 = 3 (the given starting value) Jump to any term at once. Great for a far-off term: a_ 50 = 101 in a single step. Shows how the list grows. To reach the 50th term you must build all 49 before it. You can now read a sequence's terms by position and move between the two ways of defining one. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Algebra 2 course.