A rule that turns a position number into a term

What to assume they know

The natural numbers as an ordered counting set, and the idea of a subset of them
Function notation: an input, an output, and the requirement that one input gives one output
Substituting a number for a letter in an algebraic expression, including expressions with a fraction bar and with a power
Signed arithmetic — multiplying three signed factors, and reading a power of −1 as a sign switch
Reading subscript notation, which the chapter uses from its second page onward

What they should be able to do

Distinguish the position number from the term standing at that position, and write both using the chapter's subscript notation
Generate the opening terms of a sequence from a closed expression for its general term, including expressions producing negative and zero terms
Generate the opening terms of a sequence from a rule that refers back to earlier terms, and explain why the terms must be produced in order
Decide whether a listed sequence runs out or continues without end, and justify the verdict from the rule rather than from how many terms are printed
Reach a distant term of a sequence by direct substitution without listing the terms in between
Explain why a sequence with no expression for its general term, such as the primes, is still a sequence
State the chapter's general definition of a sequence and identify what plays the part of the input set

Where it usually goes wrong

"If you cannot write a formula, it is not a sequence." The primes are the chapter's counterexample and they are placed exactly where this belief would form. What is required is that each position be determined, not that it be computable from an expression.
"The terms have to increase." Example 1(ii) descends and then hits zero; Exercise 8.1's fifth item flips sign at every step. Order in a sequence means the positions are ordered, not the values.
"The terms have to be different from each other." Fibonacci opens with two equal terms. Repetition is allowed; it is the positions that are distinct.
"aₙ and n are interchangeable." Students routinely answer "the 20th term is 20" or substitute the term back in as a position. Keep the input strip and the output strip visually separate for the whole video.
"Infinite means the values get arbitrarily large." The quotient chain runs forever and never reaches 10 ÷ 3. Infinite is a statement about how many positions there are.
"A backward-looking rule can be evaluated at any position directly." It cannot — Example 3 has to be walked. This is the itch that the next module scratches, when the geometric case gets a rule you can jump with.
"The first term is a₀." This book indexes from a₁ throughout, and every formula later in the chapter carries an exponent of n − 1 because of it. Fix the convention now or the next two topics will not add up.

Questions to check understanding

Produce the first four or five terms from a given expression for the general term, including expressions with a fraction bar and a power of −1
Produce a named far-off term by direct substitution, with the substitution shown
Produce terms from a backward-looking rule and then write the associated running total
Decide, with a reason, whether a described sequence is finite or infinite
Given several opening terms, propose an expression for the general term and test it at a position beyond those given
Say why a listed sequence such as the primes admits no expression, in words

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated inputs. The chapter prints answers for its numbered Examples only; every exercise value below is an input, not an answer.

The ancestor count (§8.2, p. 135 and §8.2, p. 136). A generation gap is taken as 30 years and the span asked about is 300 years. The page divides to get the number of generations, then lists the ancestors generation by generation: 2, 4, 8, 16, 32, …, 1024. Verified: 300 ÷ 30 = 10 generations, and 2¹⁰ = 1024, so the tenth listed value is consistent with doubling from 2. The chapter labels these a₁ = 2, a₂ = 4, a₃ = 8 and a₁₀ = 1024. This is the chapter's own worked instance of a finite sequence — it stops because the question stops it.
The quotient chain (§8.2, p. 136). Dividing 10 by 3 and halting at each successive step gives 3, then 3.3, then 3.33, then 3.333, and onward. The page labels a₁ = 3, a₂ = 3.3, a₃ = 3.33 and a₆ = 3.33333. Verified: the term at position n carries n − 1 threes after the point, which is what makes a₆ = 3.33333 consistent with a₁ = 3. This is the chapter's infinite example, and it is worth pointing out that every one of its terms stays below 10 ÷ 3.
The even numbers, read as a pattern (§8.2, p. 136). The page pairs each of the opening four terms with twice its position — 2 with 2 × 1 at position one, 4 with 2 × 2 at position two, 6 with 2 × 3, and 8 with 2 × 4 — then skips ahead and prints a₂₃ = 46 = 2 × 23 and a₂₄ = 48 = 2 × 24 before naming the rule aₙ = 2n. Verified: 2 × 23 = 46 and 2 × 24 = 48. The odd numbers 1, 3, 5, … are then given the rule aₙ = 2n − 1. The two skipped-ahead lines are doing real work: they are the demonstration that the rule is being tested away from the terms it was read off.
Fibonacci (§8.2, p. 136). Given as 1, 1, 2, 3, 5, 8, …. Both seed terms are set to 1. The third is written as the total of the two seeds. From position three onward the rule is aₙ = aₙ₋₂ + aₙ₋₁, stated for n > 2. Verified: 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, so the printed opening agrees with the relation. Note for production: the portrait caption on p. 135 dates Fibonacci 1175–1250, while the Historical Note on p. 150 dates him 1170–1250. The book disagrees with itself; do not show either range as settled.
The primes (§8.2, p. 137). Listed as 2, 3, 5, 7, …. The chapter states that no expression for the term at position n is available here and that such a sequence has to be pinned down in words. This is the example that makes the section's definition necessary rather than decorative.
Example 1 (p. 137). (i) aₙ = 2n + 5. (ii) aₙ = (n − 3) ÷ 4. Both are asked for their first three terms. Verified: (i) gives 7, 9, 11; (ii) gives −1/2, −1/4, 0. Part (ii) is the one to dwell on — it produces a negative term and then a zero term, so it kills the assumption that a sequence climbs.
Example 2 (p. 138). aₙ = (n − 1)(2 − n)(3 + n), asked at position 20. Verified: the three factors are 19, −18 and 23, whose product is −7866. The teaching point is that no earlier term was needed.
Example 3 (p. 138). a₁ = 1 with aₙ = aₙ₋₁ + 2 for n ≥ 2, asked for five terms. Verified: 1, 3, 5, 7, 9. Contrast with Example 2 deliberately: here position five cannot be reached without passing through positions two, three and four.
Exercise 8.1 inputs (pp. 138–139). Rules whose first five terms are wanted: n(n + 2); n ÷ (n + 1); 2ⁿ; (2n − 3) ÷ 6; (−1)ⁿ⁻¹ · 5ⁿ⁺¹; n(n² + 5) ÷ 4. Rules with one named term wanted: 4n − 3 at positions 17 and 24; n² ÷ 2ⁿ at position 7; (−1)ⁿ⁻¹n³ at position 9; n(n − 2) ÷ (n + 3) at position 20. Backward-looking rules: a₁ = 3 with aₙ = 3aₙ₋₁ + 2 for n > 1; a₁ = −1 with aₙ = aₙ₋₁ ÷ n for n ≥ 2; a₁ = a₂ = 2 with aₙ = aₙ₋₁ − 1 for n > 2. The last item takes the Fibonacci relation with both seeds equal to 1 and asks for the ratio of consecutive terms at the first five positions.

Figures to have open

A two-track diagram: an upper track of position boxes 1, 2, 3, …, n and a lower track of term boxes, with a labelled arrow standing for the rule. Everything in sections 2 and 7 hangs on the two tracks being visibly different objects. Standard schematic; the chapter draws nothing like it.
A dependency picture for the Fibonacci relation — two arrows converging on each new term — set beside a picture for Example 2 where a single arrow drops straight from the position onto the term. Standard schematic.
This chapter supplies almost no artwork. All sixteen printed pages were opened as images; the only illustration anywhere in pp. 135–150 is the oval portrait of Fibonacci on p. 135 with its two-line caption, and the chapter prints no numbered figure at all. Every visual for this topic has to be built from scratch.

Where this sits in the book

NCERT Class XI Mathematics, Chapter 8 "Sequences and Series", §8.1 Introduction (p. 135) and §8.2 Sequences (pp. 135–137)
Examples 1–3, pp. 137–138; Exercise 8.1, pp. 138–139
The chapter's Summary, p. 149, restates the definition of a sequence and the finite/infinite split in four sentences — the informal definition, the function definition with its domain, and one sentence each for the finite and the infinite case
The Historical Note, pp. 149–150, attributes the Fibonacci sequence and gives a date range for him that differs from the p. 135 caption