Testing a given list, and what "finite" versus "infinite" changes

What to assume they know

• A fixed step between neighbours is the whole definition — the definition of an AP as a list with one unchanging directed step, and how to read that step off neighbouring entries
• Subtracting signed numbers, fractions and decimals
• Simplifying surds far enough to see that √8 is 2√2 and √18 is 3√2
• Recognising repeated multiplication (a fixed percentage, a fixed factor) as a different rule from repeated addition

What they should be able to do

• Apply the AP test to a given list by comparing every available pair of neighbouring differences, not just the first pair
• Explain why one mismatch is conclusive while several matches are not
• Identify from a described situation whether the quantities it generates form an AP, and say which rule generates them if they do not
• Classify a printed AP as finite or infinite, and name the last term of a finite one
• Explain what the trailing dots after a list are asserting
• Justify, for surd and power lists, whether the differences are constant, by simplifying before comparing
• Recover the common difference from an AP using a single neighbouring pair, and state the condition under which that shortcut is legitimate

Where it usually goes wrong

• "Check the first two terms and you are done." The chapter's own Example 2 (iv) begins 1, 1, 1 — two perfect agreements — and then breaks. Any test that stops early passes this list, and the list is not an AP.
• "The chapter says one difference is enough, so one difference is enough." It says that about a list you have already established is an AP, where the question is only what d equals. Using it as an entry test is a misreading of its scope.
• "If I check all the printed terms and they agree, it is an AP." For a list printed with trailing dots you have checked a sample. The dots are a claim about a rule you have not been shown.
• "Terms that look alike make an AP." Four squares are not an AP; four particular surds are. Appearance is not the test; subtraction is.
• "Infinite just means very long." It means there is no last entry, which is precisely the property the sum formula later needs and cannot get from such a list.
• "A finite AP is a different object." It is the same rule stopped early. What changes is that a last entry now exists to be named and used.
• "Compound interest grows by a fixed amount." It grows by a fixed percentage, so the amount added rises every year. The chapter puts this situation in the same question as the taxi fare so the two rules can be told apart.

Questions to check understanding

• Decide which of several given lists are APs, and for each one that is, state the step and extend the list by three entries — the shape Exercise 5.1 question 4 takes, and the commonest board phrasing
• Justify why a described situation does or does not generate an AP
• Given a list of surds or powers, simplify and then decide
• State whether a given AP is finite or infinite and name its last term where there is one
• Spot the odd entry: given a list that is nearly an AP, identify which entry breaks it and what it should have been
• Reason about a list containing a letter, and state any condition on that letter needed for the list to be an AP

Examples worth working on the board

Inputs only. Values marked verified are worked out here on the chapter's printed data.

• Example 2 (pp. 54–55), four lists to be tested. Hand over the lists, not the verdicts: (i) 4, 10, 16, 22, … (ii) 1, −1, −3, −5, … (iii) −2, 2, −2, 2, −2, … (iv) 1, 1, 1, 2, 2, 2, 3, 3, 3, … Verified: (i) differences 6, 6, 6 — an AP, continuing 28 and 34. (ii) differences −2, −2, −2 — an AP, continuing −7 and −9. (iii) differences 4 then −4 — the very first comparison already fails. (iv) differences 0, 0, then 1 — two agreements followed by a disagreement. Part (iv) is the whole argument of section 4 and should be worked slowly, one difference at a time, so the moment of failure lands.
• The chapter's finite catalogue (§5.2, p. 52). Heights 147, 148, …, 157; temperatures −3.1 up to −2.5 in steps of 0.1 across seven readings; loan balances 950, 900, 850, 800, …, 50; cash prizes 200, 250, 300, 350, …, 750; savings totals 50, 100, …, 500 over ten months. Verified: eleven, seven, nineteen, twelve and ten entries respectively, each with a last entry printed. The prize list is stated as running from Class I to Class XII, which is what fixes it at twelve.
• The chapter's infinite catalogue (§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, … All five end in dots and none carries a last entry.
• The list the chapter rejects mid-section (p. 53): 1, 1, 2, 3, 5, … Verified: differences 0, 1, 1, 2 — not constant. This is the rabbit list from the introduction, returning as a counterexample.
• Exercise 5.1 question 1 (p. 55), four described situations: a taxi fare of ₹15 for the first kilometre and ₹8 for each one after; a vacuum pump that removes one quarter of whatever air remains at each stroke; a well costing ₹150 for the first metre with ₹50 more for each metre after; ₹10000 deposited at 8% per annum compound interest. Verified: the fare list 15, 23, 31, 39, … and the digging-cost list 150, 200, 250, 300, … are APs with steps 8 and 50; the pump multiplies what is left by 3/4 each time and the deposit multiplies by 1.08 each year, so neither is an AP.
• Exercise 5.1 question 4 (pp. 55–56), fifteen lists. The two planted pairs are the teaching material:
  • (xii) √2, √8, √18, √32 against (xiii) √3, √6, √9, √12
  • (xiv) 1², 3², 5², 7² against (xv) 1², 5², 7², 73 Verified: (xii) simplifies to √2, 2√2, 3√2, 4√2, an AP with step √2; (xiii) simplifies to √3, √6, 3, 2√3, whose first two differences are about 0.717 and 0.551 — not an AP. (xiv) is 1, 9, 25, 49 with differences 8, 16, 24 — not an AP; (xv) is 1, 25, 49, 73 with differences 24, 24, 24 — an AP. So in each pair the list that looks more systematic is the one that fails. Do not tell the student the verdicts first; make them subtract. The remaining items are 2, 4, 8, 16; 2, 5/2, 3, 7/2; −1.2, −3.2, −5.2, −7.2; −10, −6, −2, 2; 3, 3+√2, 3+2√2, 3+3√2; 0.2, 0.22, 0.222, 0.2222; 0, −4, −8, −12; −1/2, −1/2, −1/2, −1/2; 1, 3, 9, 27; a, 2a, 3a, 4a; a, a², a³, a⁴. Verified: the APs among these are 2, 5/2, 3, 7/2 (step 1/2); −1.2, −3.2, −5.2, −7.2 (step −2); −10, −6, −2, 2 (step 4); the 3 + k√2 list (step √2); 0, −4, −8, −12 (step −4); the constant −1/2 list (step 0); and a, 2a, 3a, 4a (step a). The powers list a, a², a³, a⁴ has differences a(a−1), a²(a−1), a³(a−1), which agree only in the degenerate cases where a is 0 or 1 — a good place to say that a list with a letter in it may need a condition attached.
• Exercise 5.1 question 3 (p. 55): read the first term and the step from 3, 1, −1, −3; from −5, −1, 3, 7; from 1/3, 5/3, 9/3, 13/3; from 0.6, 1.7, 2.8, 3.9. Verified: the steps are −2, 4, 4/3 and 1.1.

Figures to have open

• A difference strip: the list on one line, the subtractions bracketed beneath it, and a mismatch highlighted the moment it appears. Standard schematic; it is the workhorse of the whole topic.
• A finite-versus-infinite panel: one list terminated with a boxed final entry, one running off the right edge under trailing dots. Standard schematic.
• A simplification table for the surd pair — √8 → 2√2, √18 → 3√2, √32 → 4√2, and √6 and √12 left as they are. Standard schematic; the point is that you cannot compare before you simplify.
• No textbook figure is needed for this topic. Figures 5.1, 5.2 and 5.3 belong to the previous topic; the exercise pages here (pp. 55–56) carry no artwork, which I confirmed on the printed pages.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, Chapter 5 "Arithmetic Progressions", §5.2, pp. 52–55 — the finite and infinite paragraphs on p. 52, the difference computations on p. 53, the subtraction-order note and Example 2 on pp. 54–55
• Exercise 5.1, pp. 55–56, questions 1 to 4
• Within-chapter forward pointer: the last term is denoted l on p. 58 and is used in the alternative sum formula on p. 64, which is what makes the finite case matter later