A fixed step between neighbours is the whole definition

What to assume they know

• Adding and subtracting integers, decimals and fractions, including subtracting a larger number from a smaller one
• Reading a subscripted symbol such as a₃ as "the third one of these"
• Substituting a number for a letter in an expression
• That multiplying by a fixed factor and adding a fixed amount are different operations — carried over from percentage and compound-interest work in earlier classes

What they should be able to do

• State what has to be true of a list of numbers for it to be called an arithmetic progression, in terms of the difference between neighbouring entries
• Compute the common difference of a given AP by subtracting a term from the term that follows it, and explain why the subtraction runs in that order
• Give an AP with a negative common difference and one with a zero common difference, and justify both against the definition
• Write out the first four entries of an AP from a stated first term and common difference
• Write an AP in its general form and say what the coefficient of d counts in each slot
• Distinguish a list built by repeated addition from one built by repeated multiplication, using the chapter's own opening examples
• Show that the middle one of three consecutive AP entries is the average of its two neighbours, and connect that to the printed arithmetic-mean remark

Where it usually goes wrong

• "AP means the numbers get bigger." The chapter's ladder, its loan balances and its temperature-free list 100, 70, 40, 10 all decrease, and 3, 3, 3, 3 does nothing at all. The definition constrains the step, not the direction.
• "A fixed rate of increase makes an AP." The savings scheme grows by a fixed 25% each period and is not an AP — the amount added grows every time. Fixed factor and fixed addend are different rules, and the chapter puts them side by side on purpose.
• "Subtract the smaller from the larger." Doing that to 6, 3, 0, −3 gives a step of +3 and the wrong list. The step is always the later term minus the earlier one, sign included; the chapter flags this in its own words on p. 54.
• "Zero cannot be a common difference." A list of identical numbers meets the definition exactly, and the chapter includes 3, 3, 3, 3 and the pair (2, 0) precisely so this case is on the record.
• "Knowing the first term tells you the AP." From a = 6 alone you cannot choose between 6, 9, 12, … and 6, 3, 0, … The chapter asks this question directly and answers that both numbers are needed.
• "The squares 1, 4, 9, 16 are an AP because they follow a clear pattern." A pattern is not the same as a constant step. The differences 3, 5, 7 are themselves an AP, which is a different and later idea.
• "a + 2d is the second term." The multiplier counts the steps taken, not the position reached. This is the misreading that makes the nth-term rule look arbitrary in the next module.

Questions to check understanding

• Given a list, state whether it is an AP and give the common difference
• Given a situation in words (a fare, a wage, a repayment schedule), decide whether the quantities form an AP and justify the answer
• Write the first four terms from a given first term and common difference, including cases where the difference is negative, fractional or zero
• Given the first two terms, write the next three
• Find the missing entry in a short AP by using the equal-step property
• Show that three given expressions are in AP by checking that the middle one is the average of the outer two

Examples worth working on the board

Inputs only — the explanation system derives the rest. Values marked verified are worked out here on the chapter's printed data.

• The six opening patterns (§5.1, pp. 49–50). Hand all six over as raw data so the sorting in section 2 has something to sort:
  • a starting monthly salary of ₹8000 with ₹500 added each year, giving 8000, 8500, 9000, …
  • the eight rung lengths of the ladder in Fig. 5.1, in cm from the bottom up: 45, 43, 41, 39, 37, 35, 33, 31
  • a savings scheme in which the amount becomes 5/4 of itself every three years; ₹8000 matures to 10000, 12500, 15625, 19531.25 after 3, 6, 9 and 12 years
  • the number of unit squares in squares of side 1, 2, 3 units — 1², 2², 3², … (Fig. 5.2)
  • ₹100 into a money box on the first birthday, increased by ₹50 each year: 100, 150, 200, 250, …
  • the rabbit-pair counts 1, 1, 2, 3, 5, 8 over six months (Fig. 5.3) Verified: the salary list steps by +500, the rung list by −2, the money-box list by +50 — three APs. The savings scheme multiplies by 1.25 each time (10000 → 12500 → 15625 → 19531.25), the squares step by 3 then 5 then 7, and the rabbit counts step by 0, 1, 1, 2, 3 — none of these three is an AP.
• The five lists of §5.2 (p. 51): 1, 2, 3, 4, …; 100, 70, 40, 10, …; −3, −2, −1, 0, …; 3, 3, 3, 3, …; −1.0, −1.5, −2.0, −2.5, … Verified: the common differences are 1, −30, 1, 0 and −0.5. The third crosses zero and the fourth never moves; both still satisfy the definition.
• Five further examples the chapter lists but leaves unexplained (§5.2, p. 52): heights in cm 147, 148, …, 157; a week of minimum temperatures in °C −3.1, −3.0, −2.9, −2.8, −2.7, −2.6, −2.5; loan balances in ₹ after paying 5% of ₹1000 monthly, 950, 900, 850, 800, …, 50; cash prizes in ₹ 200, 250, 300, 350, …, 750; monthly savings totals in ₹ 50, 100, …, 500. Verified: the common differences are 1, 0.1, −50, 50 and 50. The chapter explicitly leaves the justification to the reader.
• **Building an AP from a and *d*** (§5.2, pp. 52–53). The chapter runs six pairs: (6, 3); (6, −3); (−7, −2); (1.0, 0.1); (0, 1½); (2, 0). Verified: these generate 6, 9, 12, 15, …; 6, 3, 0, −3, …; −7, −9, −11, −13, …; 1.0, 1.1, 1.2, 1.3, …; 0, 1½, 3, 4½, 6, …; and 2, 2, 2, 2, …
• **Reading d back off a list** (§5.2, p. 53). For 6, 9, 12, 15 the successive differences are 3, 3, 3; for 6, 3, 0, −3 they are −3, −3, −3. Verified. Note the second list is the first one's mirror and shows the sign carrying the direction.
• The subtraction-order warning (§5.2, p. 54). For 6, 3, 0, −3 the step is found as 3 − 6, not 6 − 3. Verified: reversing it gives +3, which would predict 9 as the second term and contradict the list in front of you.
• Example 1 (p. 54): the AP 3/2, 1/2, −1/2, −3/2, … Verified: the first term is 3/2 and the step is 1/2 − 3/2 = −1. The terms pass through zero's neighbourhood without any special case arising.
• The arithmetic-mean remark (p. 72). Three numbers in AP satisfy b = (a + c)/2. Verified as a derivation: equal steps means b − a = c − b, so 2b = a + c.

Figures to have open

• Fig. 5.1, the ladder (p. 49). Its eight rungs are drawn in perspective, narrowing from bottom to top, and I confirmed on the artwork that the drawing carries no numbers at all — the eight lengths live only in the running text. Redraw it as a schematic with the eight lengths written against the rungs, since the whole point is that consecutive rungs differ by the same 2 cm.
• Fig. 5.2, the three squares of side 1, 2 and 3 with their unit cells drawn in (p. 50). Verified on the printed page: exactly three squares, shaded, showing 1, 4 and 9 cells. Needed as the counterexample in section 2.
• A number line carrying a growing AP, a shrinking AP and a constant AP with equal-length arrows between consecutive marks. Standard schematic; the equal arrow lengths are the argument.
• A two-row strip: a, a + d, a + 2d, a + 3d on top and the step count 0, 1, 2, 3 underneath. Standard schematic, and it sets up the next topic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, Chapter 5 "Arithmetic Progressions", §5.1 Introduction, pp. 49–51, and §5.2 Arithmetic Progressions, pp. 51–55 up to Example 1 on p. 54
• Figures used: Fig. 5.1 (p. 49), Fig. 5.2 (p. 50), Fig. 5.3 (p. 50)
• Within-chapter forward pointer: the A Note to the Reader box on p. 72, which states the arithmetic-mean relation; it sits after §5.5 and is set in its own tinted box, separate from the summary list
• Exercise 5.1 items 2 and 3 (p. 55) drill exactly the two directions covered here — build a list from a and d, and read a and d off a list