An explicit rule computes any term straight from n

What to assume they know

• Sequences, terms and subscript notation (Order is the point: a sequence carries position as well as value)
• Substituting a number for a letter in an expression, and evaluating it (Turning a situation into an expression: terms, variables, coefficients)
• Solving a one-step linear equation in one unknown (What makes a polynomial linear, and the equation you get by fixing its value)
• Knowing which numbers are counting numbers, and that 94.6 is not one
• Prime numbers and how they are recognised

What they should be able to do

• Read an expression in n as a rule for the term in position n, and generate the first several terms from it
• Compute a distant term of a sequence directly, without listing the terms before it
• Test whether a given number is a term of a sequence by solving the rule as an equation
• Interpret a non-integer solution as proof that the candidate is not a term, and explain why
• Report the position of a term as well as its value
• Recognise that some sequences, the primes among them, come with no simple explicit rule

Where it usually goes wrong

• "You still have to know the earlier terms." You do not, and that is the whole point. Ask for term 1170 and let the two-step arithmetic finish before a student walking the list has reached term 20.
• "If solving gives a decimal, I should round it." Rounding here manufactures a false answer. A decimal position means the candidate sits between two terms, so it is not a term at all. 94.6 is not "about the 95th term" — the 95th term is 473.
• "Any number bigger than the first term must be somewhere in the list." 5n − 2 produces 3, 8, 13, 18, … and skips everything in between. Being large enough is not the same as being reachable.
• "'Is 557 a term?' is a trick question, so the answer is no." In the p. 178 exercise it is yes, at position 188. The method decides; the phrasing does not.
• "A rule that matches the first few terms is the rule." The primes start 2, 3, 5, 7 and any number of rules match those four. Matching a prefix is evidence, not proof — which is exactly why the chapter asks about the primes and then drops the question.
• "Every sequence has an explicit rule." The chapter's own primes example and its opening remark in §8.4 say otherwise.
• "Position and value are interchangeable." Solving for n returns a position, and the answer to "which term is 137?" is 69, not 137. Make students say which of the two numbers they have found.

Questions to check understanding

• Generate the first five terms from a given rule in n
• Find a specified distant term of a sequence given its rule
• Decide whether a stated number is a term of a given sequence, showing the equation and interpreting the solution
• Name the position of a given term
• Given the rule, find which term equals a stated value — the board's standard "which term of the sequence is …?" phrasing
• Explain in words why a fractional solution rules the candidate out
• Explain why the first ten primes do not settle a rule for the eleventh

Examples worth working on the board

Values marked verified are worked out here; the chapter states some of them and leaves the rest as questions.

• Example 1 (§8.2, p. 177). The rule uₙ = 2n − 1. The page substitutes 1, 2 and 3 and prints u₁ = 1, u₂ = 3, u₃ = 5, then identifies the result as the odd numbers.
• The jumping exercise (§8.2, p. 177). Using the same rule, find the terms in positions 53, 108 and 1170. Verified: 105, 215, 2339. The page also mentions positions 20, 53 and 300 as the kind of thing a rule makes easy. The explanation should stress that all three cost one multiplication and one subtraction.
• Locating 137 (§8.2, p. 177). Solve 2n − 1 = 137. The page prints n = 69 and reports 137 as the 69th odd number. Verified: 2 × 69 − 1 = 137.
• Example 2 (§8.2, p. 177). The rule sₙ = 5n − 2. The page asks for the first six terms and for the terms in positions 100 and 1000. Verified: the first six are 3, 8, 13, 18, 23, 28; the 100th is 498; the 1000th is 4998.
• Membership, worked (§8.2, p. 177). Solving 5n − 2 = 308 gives 5n = 310 and n = 62, so 308 is the 62nd term. Solving 5n − 2 = 471 gives 5n = 473 and n = 94.6, which is not a counting number, so 471 is not a term. The page then asks the student to explain why the position has to be a counting number — that question is section 9's job.
• A printed inconsistency to steer around (§8.2, p. 177, verified on the printed page). The sentence that sets the task announces two candidates, 308 and 473, and the working that follows tests 308 and then 471. The arithmetic shown is correct for 471. But 473 is a term: solving 5n − 2 = 473 gives n = 95, so it is the 95th term.
• The exercise on a third rule (§8.2, p. 178). For tₙ = 3n − 7: find the terms in positions 1, 2, 3, 12, 18 and 50; find which position holds 332; and decide whether 557 occurs. Verified: the terms are −4, −1, 2, 29, 47, 143; 332 = 3n − 7 gives n = 113; 557 gives 3n = 564 and n = 188, so 557 is a term. Note as a check: unlike the 471 case, this rejection candidate does not reject — the exercise's third part is a "yes, and here is where" question, and an explanation that assumes "is it a term?" always means "no" gets it backwards. This rule also produces negative terms early, which is worth showing.
• The primes (§8.2, p. 177). The page prints the first ten: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and asks whether a rule can be found that predicts the next few. It offers none. §8.4 opens by saying outright that this list has no clear regularity (p. 180), so the honest answer to the p. 177 question is no rule of the kind this chapter builds.
• A useful contrast for section 3 (§8.3, p. 178, a deliberate forward reference): 1, 4, 7, 10, 13, … has the explicit rule tₙ = 3n − 2. Having three rules of the same shape on hand — 2n − 1, 5n − 2, 3n − 2 — lets the explanation show that the multiplier is the gap and the constant sets the start.

Figures to have open

• A rule machine: a box labelled with the expression, a slot for n on the left and the term coming out on the right. Standard schematic, reused from the chapter's own framing of a rule as something you substitute into.
• A position axis carrying only whole ticks, with 94.6 marked in the empty space between 94 and 95, and the terms 468 and 473 sitting on those two ticks. Standard schematic; this is the figure the chapter does not draw and section 9 needs.
• A three-column table of the chapter's three rules of this shape — 2n − 1, 5n − 2, 3n − 2 — with their first four terms, so the multiplier and the constant can be read off. Standard schematic.
• No figure from the textbook is required; §8.2 carries no numbered figure.

Where this sits in the book

• Chapter 8, §8.2 Explicit Rule for a Sequence, pp. 176–178. The section's definition sentence sits at the foot of p. 176; Examples 1 and 2 are on p. 177; the closing exercise runs onto p. 178.
• Two Think and Reflect boxes belong here: why an explicit rule is worth having (p. 177) and finding the rule for the square numbers (p. 177). The second is answered by tₙ = n², which the chapter does not print in §8.2.
• Deliberate cross-references outside this topic: §8.3 (p. 178) supplies the recursive alternative; §8.4 (p. 180) is where the chapter states that the primes follow no clear rule; the summary restates the explicit-formula definition on p. 196.
• Exercise Set 8.1, p. 179, items 1–4 are the assessment for this section: three rules to generate from, the 10th and 15th terms of 5n − 3, whether 97 and 172 occur in it, and which of its terms is 607.