Building the rule for the nth term out of repeated addition

What to assume they know

• A fixed step between neighbours is the whole definition — an AP is fixed by its first entry and its unchanging directed step
• Testing a given list, and what "finite" versus "infinite" changes — how to confirm from a given list that the step really is constant
• Collecting repeated addition into multiplication: five lots of 500 is 5 × 500
• Substituting a number for a letter and evaluating with a signed multiplier

What they should be able to do

• Extend an AP one step at a time and describe why that method is impractical for a distant entry
• Rewrite a chain of identical additions as a single multiplication and identify what the multiplier counts
• Derive the nth-term rule by inspecting the second, third and fourth entries written in terms of the first entry and the step
• Explain in words why the step count is n − 1 rather than n
• Apply the rule to compute a stated entry of a given AP
• Use the general term to describe an entry of an AP without computing it
• Recognise the notation for a last entry of a finite AP and say what letter stands in for it

Where it usually goes wrong

• "The nth entry is a + nd." This is the single most common error on the topic. It gives the entry one position further on. Anchor the correction to the salary: a + 5 × 500 is what she earns in her sixth year, not her fifth.
• "The (n − 1) is a rule you memorise." It is a count of additions performed. The first entry is where you stand before you move, so it costs nothing; every entry after it costs one step.
• "The formula is a shortcut, so the long way must be wrong." The long way is correct and is how the rule was found. It is only slow.
• "aₙ is a different quantity from the terms in the list." It is the same list, addressed by position instead of by writing everything before it.
• "The general term only works for whole positions." Positions are counted, so n is a positive whole number. The expression will happily accept 2.5 and return a number that is not on the list — a point the next topic turns into a test.
• "You need to know the whole list to find a distant entry." Two numbers and a position are sufficient. That is the payoff of the previous module's claim that a and d carry all the information.

Questions to check understanding

• Find a stated entry of an AP given its opening entries
• Find a stated entry given the first entry and the step directly
• Write the general term of an AP described in words
• Given the general term, produce the first few entries and the step
• Word problems that run the salary structure again with different numbers — a starting wage and a fixed yearly rise, asked for a distant year
• Explain, in one line, why the step is multiplied by n − 1

Examples worth working on the board

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

• The salary situation (§5.1 item (i), p. 49; reopened as §5.3, p. 56). Starting monthly salary ₹8000; annual increment ₹500. The chapter asks for the fifth year's salary. Verified: second year 8500; third 9000; fourth 9500; fifth 10000.
• The collapse, written out (§5.3, p. 56). The chapter shows the third year four ways in succession: 8500 + 500, then 8000 + 500 + 500, then 8000 + 2 × 500, then 8000 + (3 − 1) × 500. It repeats the identical four-line treatment for the fourth and fifth years. Verified: 8000 + 2 × 500 = 9000; 8000 + 3 × 500 = 9500; 8000 + 4 × 500 = 10000. This staircase is the derivation.
• The fifteenth year (§5.3, p. 57). The printed working shows the salary for the fourteenth year plus one more ₹500, then a bracket containing thirteen copies of 500 marked with an underbrace reading 13 times, then 8000 + 14 × 500. Verified: 13 + 1 = 14 additions in total, and 8000 + 14 × 500 = 15000. The underbrace is worth working through — it is the moment the count and the position are visibly one apart.
• The twenty-fifth year (§5.3, p. 57). Verified: 8000 + 24 × 500 = 8000 + 12000 = 20000.
• The derivation in letters (§5.3, p. 57). Take a first entry a and a step d. The second entry sits one step along, so it carries a single copy of the step, and the chapter writes that multiplier as (2 − 1). Take another step and the third entry carries two copies, multiplier (3 − 1). Once more and the fourth carries three, multiplier (4 − 1). Verified as an argument: each line gains exactly one further copy of the step over the line above it, and every multiplier lands one below its own position, so the nth line is forced to read a + (n − 1)d.
• Example 3 (p. 58): the AP 2, 7, 12, … and its tenth entry. Verified: a = 2, d = 5, so the tenth entry is 2 + 9 × 5 = 47. Note that 9, not 10, multiplies the step.
• A contrast. Take the same first entry and step and compute both a + nd and a + (n − 1)d for n = 5: 8000 + 5 × 500 = 10500 against 8000 + 4 × 500 = 10000. Verified. The first answer is the sixth year's salary, not the fifth — the off-by-one has a concrete meaning, and showing it as a wrong year rather than a wrong number is what makes it stick.
• Notation (§5.3, p. 58). Where an AP has m entries, the chapter writes the final one as a\_m and notes that a last entry is sometimes written l instead. Caution for whoever draws the slides: in this typeface the italic l is easy to mistake for the digit 1, and it does render that way on p. 64.

Figures to have open

• A year strip: positions 1 to 6 along the top, cumulative step counts 0 to 5 along the bottom, and the running salary above each position. Standard schematic, and it carries sections 4, 5 and 8 on its own.
• The underbrace panel from p. 57 redrawn: a bracket over thirteen copies of the increment, labelled with the count, and the fourteenth increment standing apart. This is the chapter's own working.
• A stack of the four lines a₁, a₂, a₃, a₄ with the multipliers 0, 1, 2, 3 aligned in a visible column. Standard schematic; the column is the argument.
• No textbook figure is required. Fig. 5.1 belongs to §5.1 and this section prints no artwork of its own.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, Chapter 5 "Arithmetic Progressions", §5.3 nth Term of an AP, pp. 56–58 up to Example 3
• The salary situation this section reopens is stated in §5.1 item (i), p. 49
• Exercise 5.2 question 1 (p. 61) and question 2 (p. 62) are the direct drill on this rule; the remaining Exercise 5.2 items belong to the next topic
• The rule is restated as point 3 of the chapter summary, §5.5, p. 72