Common ratio, and why the nth term is ar^(n−1)

What to assume they know

• Sequences, terms and subscript notation (Order is the point: a sequence carries position as well as value)
• Explicit and recursive rules (An explicit rule computes any term straight from n, A recursive rule builds each term from the ones before it)
• The arithmetic progression and its (n − 1) step count (Common difference, and why the nth term is a + (n − 1)d)
• Powers with whole-number exponents, and that any non-zero base to the power 0 is 1
• Dividing one fraction by another (What "rational" means, and why the denominator cannot be zero)
• Multiplication of negative numbers (Why a debt times a debt is a fortune)

What they should be able to do

• Test a sequence for a constant ratio and decide whether it is a geometric progression
• Identify the first term and the common ratio of a given GP
• Derive tₙ = arⁿ⁻¹ by counting the multiplications between the first term and the nth
• Explain why the exponent is n − 1 rather than n
• Write both the explicit and the recursive rule for a given GP
• Handle common ratios that are whole, fractional, negative and irrational
• Find which term of a GP equals a stated value, by matching powers
• Contrast a GP with an AP on the same data and say which test settles it

Where it usually goes wrong

• **"The nth term is arⁿ."** Test it on the first term: that gives ar, one multiplication too many. Standing on the first term you have multiplied by nothing.
• "A geometric progression grows." Example 9's ratio is 3/4 and its terms fall; Example 8's ratio is −1 and its terms alternate between two values forever. Both are GPs. Growth is a consequence of r being bigger than 1, not part of the definition.
• "Look at the differences." For 3, 6, 12, 24 the differences are 3, 6, 12 — they are not constant, and a student who only knows the AP test concludes there is no pattern. The chapter's Think and Reflect on p. 186 is engineered to trigger exactly that moment; use it.
• "The ratio only has to work once." Every consecutive pair must give the same value. The page divides five pairs for a reason; 1, 2, 4, 7 passes the first test and fails the second.
• "A negative ratio means the terms get smaller." With r = −2 the terms grow in size while flipping sign. Size and sign are separate consequences of r.
• "An irrational ratio is not allowed." Exercise Set 8.3 item 6 uses √2. Nothing in the definition restricts r to fractions.
• "An AP with a big common difference beats a GP." Only for a while. Put 1, 5, 9, 13 beside 3, 6, 12, 24 and let the second overtake — the exponent wins eventually, whatever the difference is.
• **"r can be anything at all."** A ratio of 0 would collapse every later term to zero and there would be nothing to divide by; the chapter's examples never use it.

Questions to check understanding

• Decide whether a given sequence is a GP and state a and r
• Find a stated distant term of a GP, including from another term and the ratio
• Find which term of a GP equals a given value, by matching powers
• Write both the explicit and the recursive rule for a given GP
• Find a GP from two conditions on its terms, and report both answers when the ratio turns out to have two signs
• Model a doubling or fixed-percentage situation as a GP
• Given a sequence, say which test — difference or ratio — settles what it is, and carry it out

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated data; this book prints no answer key.

• Fig. 8.6 (§8.6, p. 186). Four stages of a tile pattern, labelled Stage 1 to Stage 4, each drawn as a block of pale green squares outlined in pink. Read off the page: every stage is three squares wide, and the number of rows doubles — one row, then two, then four, then eight. So the counts are 3, 6, 12, 24, which the page states. The Think and Reflect beside it asks for stages 5 and 6, 10, 11 and 12, stage 20, and any stage, and then — the pointed question — how this differs from the pattern of Fig. 8.3.
• The additive description, and its limits (p. 186). The page first writes the counts as 3, then 3 + 3 = 6, then 6 + 6 = 12, then 12 + 12 = 24. Every step is an addition, but the amount added changes each time, so this description yields no rule. That failure is what motivates the next line.
• The multiplicative rewrite (§8.6, p. 187). The same counts become 3, 3 × 2, 3 × 4, 3 × 8, then 3, 3 × 2, 3 × 2², 3 × 2³, giving tₙ = 3 × 2ⁿ⁻¹. The page also prints the step rule: t₁ = 3 with tₙ = 2tₙ₋₁ for n ≥ 2.
• The first six terms (p. 187): 3, 6, 12, 24, 48, 96.
• The ratio test, as printed (p. 187): the page sets out 6/3, 12/6, 24/12, 48/24 and 96/48 and states that all five equal 2. This is the figure to show — five divisions, one answer.
• The general form (p. 187): a, ar, ar², ar³, …, arⁿ⁻¹.
• Examples 6, 7 and 8 (p. 187), three one-line verdicts the page asks for and does not answer. 1, 2, 4, 8, 16, …; 1, 3, 9, 27, 81, …; 1, −1, 1, −1, 1, …. Verified: all three are GPs, with ratios 2, 3 and −1. The third is the important one — it neither grows nor shrinks, and it is still a GP.
• Example 9, worked in full on the page (pp. 187–188). The sequence 5, 15/4, 45/16, 135/64, …. The page divides each term by the one before it, showing 15/4 ÷ 5 = 3/4, then 45/16 ÷ 15/4 = 3/4, then 135/64 ÷ 45/16 = 3/4, concludes that a = 5 and r = 3/4, and gives the nth term as 5 × (3/4)ⁿ⁻¹. This is the chapter's only fully worked ratio test, and it is worked on fractions on purpose: the ratio is below 1, so the sequence falls.
• Three sequences to classify (the exercise on p. 188): 2, 10, 50, 250, …; 4, 8/3, 16/9, 32/27, …; 3, −3/2, 3/4, −3/8, …. Verified: all three are GPs with ratios 5, 2/3 and −1/2, so the nth terms are 2 × 5ⁿ⁻¹, 4 × (2/3)ⁿ⁻¹ and 3 × (−1/2)ⁿ⁻¹. The third alternates in sign and shrinks, which is the case students find hardest to accept.
• A recursive rule to find (the second exercise on p. 188): give a step rule for the sequence 3, 30, 300, 3000, … whose explicit rule is 3 × 10ⁿ⁻¹. Verified: t₁ = 3 with tₙ = 10tₙ₋₁ for n ≥ 2.
• Exercise Set 8.3, items 1, 2, 4 and 6 (pp. 193–194). Verified: a GP with ratio 2 whose 8th term is 192 has 12th term 192 × 2⁴ = 3072 — four steps, four multiplications, and no need to find a. The GP 5, 25, 125, … has tₙ = 5ⁿ and tenth term 9765625. In 2, 6, 18, … the value 4374 sits in position 8, since 4374/2 = 2187 = 3⁷. In 2, 2√2, 4, … the ratio is √2 and 128 sits in position 13, since 128 = 2⁷ and 2 × (√2)¹² = 2 × 2⁶. That last item is the only irrational ratio in the chapter and is worth its own beat.
• End-of-chapter items (pp. 195, all starred). Problem 7: bacteria doubling every hour from 30 originally; how many at the end of hour 2, hour 4, and hour n? Verified: 120, 480, and 30 × 2ⁿ. Problem 10: which term of 2, 8, 32, … is 131072? Verified: position 9, with explicit rule 2 × 4ⁿ⁻¹ and step rule t₁ = 2, tₙ = 4tₙ₋₁. Problem 5: a GP whose first two terms total −4 and whose fifth term is four times its third. Verified: r² = 4, so r = 2 with a = −4/3, or r = −2 with a = 4 — two answers, which is the lesson. Problem 13: first three terms totalling 26 with squares totalling 364. Verified: 2, 6, 18.
• The AP contrast to keep in view in section 10. Pick the pair so that the GP starts behind and has to catch up, or the figure has nothing to show. The taxi fare of Example 5 (p. 183) does that job against this section's tile GP: the fare runs 240, 280, 320, …, 40 more each time, while the tiles run 3, 6, 12, 24, …. Verified: the fare leads all the way to position 8 (520 against 384) and the tiles pass it at position 9 (560 against 768), after which the gap never closes again — 600 against 1536 by position 10. Both sequences are the chapter's own. Do not use the tile AP 1, 5, 9, 13, 17, 21 for this: the GP 3, 6, 12, 24 is already ahead of it at the first position and stays ahead, so there is no crossing to show and a common difference of 4 is not the "big" one the misconception needs.

Figures to have open

• Fig. 8.6 redrawn as a schematic: four blocks, all three squares wide, with 1, 2, 4 and 8 rows, and the newly added rows shaded so the doubling is visible rather than asserted. Must keep the counts 3, 6, 12, 24. The printed art is a painted panel and need not be reproduced.
• A multiplication-counting strip: the terms of a general GP in a row with the multiplications numbered 0, 1, 2, 3 beneath the gaps. This is the figure the chapter does not draw and section 4 depends on. Standard schematic, and a deliberate twin of the addition-counting strip in Common difference, and why the nth term is a + (n − 1)d.
• A ratio-test panel: five fractions of consecutive terms, all reducing to the same value, with the fraction working shown for Example 9. Standard schematic.
• A race chart for section 10 with the AP and the GP plotted against position through position 10. Standard schematic; the chapter's own plots are Figs. 8.4 and 8.9 and are treated in An AP plots as points on a straight line and A GP plots as a curve, and what that curve tells you.

Where this sits in the book

• Chapter 8, §8.6 Geometric Progressions, pp. 186–188. Fig. 8.6 and the additive description are on p. 186; the powers rewrite, the ratio list, the general form and Examples 6 to 9 run from p. 187 to the top of p. 188; the two inline exercises close p. 188.
• The Think and Reflect on p. 186, which is the section's hinge because it asks how this pattern differs from the arithmetic one of Fig. 8.3.
• Exercise Set 8.3, pp. 193–194, items 1, 2, 4 and 6. Item 3 is a recursion and belongs to A recursive rule builds each term from the ones before it; item 5 is a bouncing ball and item 7 the square carpet, which belong to A GP plots as a curve, and what that curve tells you and Fractals: self-similarity generates a GP.
• End-of-Chapter Exercises, p. 195, problems 5, 7, 10, 11, 12 and 13, all carrying a printed asterisk.
• The summary's statement of the GP definition and of arⁿ⁻¹ is on p. 196.
• Deliberate cross-reference outside this topic: §8.6.1 and §8.6.2, pp. 188–193, apply this formula to fractals and to graphs — Fractals: self-similarity generates a GP and A GP plots as a curve, and what that curve tells you.