6.4 Sequences, Series, and Recursion

Ordered Patterns, Not Just Lists

The discrete mathematics competency in AEPA Domain V includes sequences and series, including limits and recursive definitions. This topic overlaps with functions and algebra, but the discrete emphasis is on whole-number term positions and patterns generated step by step. A sequence is an ordered list of terms, often written a_1, a_2, a_3, and so on. A series is a sum of terms. The distinction is essential: the tenth term and the sum of the first ten terms are different quantities.

An arithmetic sequence has a constant difference. If a_1 = 7 and d = 4, then a_n = 7 + (n - 1)4. The first term uses n = 1, not n = 0, unless the problem defines indexing differently. A geometric sequence has a constant ratio. If a_1 = 3 and r = 2, then a_n = 3(2)^(n - 1). The phrase "increases by 20%" signals multiplication by 1.20, not addition of 20. Percent-change sequences are geometric when the percent applies repeatedly to the new value.

Explicit, Recursive, and Context Forms

A recursive definition is incomplete without a starting value. The rule a_n = a_(n-1) + 5 describes infinitely many sequences unless a_1 is specified. For example, 2, 7, 12, ... and 100, 105, 110, ... both satisfy the same recursive update. Teacher-certification questions may ask what feedback to give a student who writes only the recursion. The mathematical issue is not notation preference; it is missing initial condition.

Worked Example: Term Versus Sum

A classroom savings project starts with $40 and adds $15 each week. If week 1 is $40, the amount in week n is a_n = 40 + 15(n - 1). In week 12, the amount is 40 + 15(11) = 205. The sum of the first 12 weekly balances is not 205. For an arithmetic series, S_n = n(a_1 + a_n) / 2, so S_12 = 12(40 + 205) / 2 = 1470. A distractor that gives 205 may be correct for the term but wrong for the accumulated sum.

For a geometric example, suppose a medication amount is reduced by 30% each hour, starting at 200 mg. The multiplier is 0.70, so a_n = 200(0.70)^(n - 1) if n = 1 is the starting amount. The long-run limit is 0 because repeated multiplication by a number between 0 and 1 drives the terms toward zero. A common mistake is subtracting 30 each hour, which would be arithmetic and eventually negative, an impossible model for this context.

Series and Limits

A finite arithmetic series can be found by averaging the first and last terms and multiplying by the number of terms. A finite geometric series with first term a_1 and ratio r uses S_n = a_1(1 - r^n) / (1 - r), for r not equal to 1. An infinite geometric series converges only when |r| < 1, and its sum is a_1 / (1 - r). If |r| >= 1, the terms do not shrink toward zero, so the infinite sum does not settle to a finite value.

Sequence limits are about term behavior as n grows. The sequence a_n = 5 + 2/n approaches 5, because 2/n approaches 0. The sequence (-1)^n does not approach a single value because it alternates between -1 and 1. The sequence 3n + 2 grows without bound. AEPA items may not ask for formal epsilon language, but they can ask which graph or description matches long-run behavior.

Student Misconceptions to Diagnose

Students often treat the term number as the term value. In 4, 9, 14, 19, the fourth term is 19, not 4. They may also use n instead of n - 1 in explicit formulas, producing an off-by-one error. Another frequent issue is identifying a pattern from too few terms. The list 2, 4, 8 could be geometric doubling, but a different rule can also produce those first three terms. In teaching contexts, ask whether the rule is stated or merely conjectured.

When reading AEPA answer choices, look for these traps: adding the common ratio, multiplying by the common difference, finding a partial sum when the question asks for a term, or using an infinite-series formula when the series is finite. The fastest path is to write what a_1 means, what n means, and whether the question wants a term, a sum, a recursion, or a limit.

Finite differences provide another quick diagnostic. Constant first differences identify a linear or arithmetic pattern. Constant second differences suggest a quadratic pattern, which may appear when a context counts cumulative growth such as handshake totals or stacked-dot figures. A teacher candidate should not force every pattern into arithmetic or geometric form; the evidence in the table determines the model. When the pattern is only conjectured from a diagram, state the assumption before extending it.