Chapter 6.5 — Exercise 6.4 — Geometric Progression

Geometric Progressions: A Fixed Ratio Every Time

A geometric progression (GP) is a sequence in which every term, after the very first one, is obtained by multiplying the term immediately before it by the same fixed number — the common ratio, written r. Just as an AP's first term has no predecessor to be compared against, a GP's first term is simply given as a starting value, with the ratio only governing every term that follows it.

3, 6, 12, 24, 48 has r=2 (each successive term simply doubles). 2, 10, 50, 250, 1250 has r=5 throughout. ½, 1/8, 1/32, 1/128 has r=¼ instead. 0.2, 0.04, 0.008, 0.0016 has r=0.2. Notice how differently these grow compared to an AP with a similarly modest-looking difference: doubling five times already multiplies the starting value by 32, far outpacing anything a fixed addition could produce over the same five steps. More generally, any geometric progression always has this same recognisable form:

Three Real Situations, Tested

Just as fixed-amount growth signalled an AP in Exercise 6.1, fixed-rate growth signals a GP — and it's worth checking a situation carefully rather than assuming from a rough description, since the two patterns can sound almost identical in plain English while behaving completely differently in the actual numbers.

• Sharmila's salary: ₹5,00,000 in year one, with a 10% increase applied every following year — 500000, 550000, 605000, 665500, … Each ratio checks out to exactly 11/10. This is a GP, r = 1.1.
• Bricks per stair step: a 30-step staircase, 100 bricks needed for the bottom step, then 2 fewer bricks required for each step after that — 100, 98, 96, 94, … The ratios (49/50, 48/49, 47/48, …) are not equal to each other at all, but the differences genuinely are: every step uses exactly 2 fewer bricks than the one before it. Not a GP — this is actually an AP, d = −2, a useful reminder that "decreasing by a fixed amount" and "decreasing by a fixed rate" describe two genuinely different patterns, however similar they might sound out loud.
• Nested triangle perimeters: starting from an equilateral triangle of side 24 cm, each new triangle is formed by joining the midpoints of the one before it, shrinking every side by exactly half. Perimeters run 72, 36, 18, 9, … This is a GP, r = ½ — a shrinking GP behaves exactly like a growing one, just with a ratio smaller than 1 instead of larger.

Four GPs, Continued

The third row shows something an AP can never do: a negative common ratio makes the sequence alternate in sign every term (81, then −27, then 9, then −3, …), since multiplying by a negative number flips the sign each time. An AP with a negative d shrinks steadily in one direction; a GP with a negative r bounces back and forth.

Nine Sequences, Classified

Three sequences are worth pulling out for individual explanation rather than a table row:

• 3, −9, 27, −81, … — ratios of −9/3=−3 and 27/(−9)=−3 both confirm the same constant ratio throughout. This is a GP, r=−3, with next terms −81, 243, −729, continuing the alternating-sign pattern already seen above.
• x, 1, 1/x, … — a GP built from a variable rather than a specific fixed number. The ratio is 1/x throughout (1÷x=1/x, and (1/x)÷1=1/x), so this is a GP, r=1/x, with next terms 1/x², 1/x³, 1/x⁴ — a pattern that holds regardless of whatever actual value x eventually turns out to be.
• 1/√2, −2, 4√2, … — a genuinely square-root-based GP, distinct from the exponent-based one above despite a superficially similar look on the page. The ratio works out to −2√2 throughout, checked both ways, so this is a GP, with next terms −16, 32√2, −128.

1/2, 1/4, 1/6, … is the one sequence here worth a direct warning: it looks like it should be geometric, since every denominator grows in what seems like a clean pattern. But the ratios are (1/4)/(1/2)=1/2 and (1/6)/(1/4)=2/3 — not equal — so despite its tidy appearance, it fails the GP test and isn't actually one. 5, 55, 555, … fails for a related but distinct reason: the ratios 55/5=11 and 555/55≈10.09 aren't equal either, even though every term is built the same visual way (repeating the digit 5) — a shared-looking construction rule for a sequence is not the same thing as a shared ratio between its terms.

Finding x From Three Consecutive Terms

Given that x, x+2, and x+6 are consecutive terms of a GP, the middle term squared must equal the product of the outer two — the defining property of any three consecutive GP terms. This follows directly from the ratio test: if the three terms share a common ratio r, then (x+2)/x = (x+6)/(x+2), and cross-multiplying turns that ratio equation straight into a middle-term-squared equation.

Checking: with x=2, the three terms are 2, 4, 8 — a GP with r=2, exactly as required. This "middle squared equals the product of the outer two" shortcut is worth remembering on its own, since it settles any three-consecutive-term GP question without ever needing to compute r explicitly first.

Reaching Any Term Directly

Every GP in this exercise stopped at "here are the next three terms," found by repeated multiplication — fine for three terms, but not remotely practical for finding the fiftieth without a formula. Exercise 6.5 derives the GP equivalent of the AP formula from Exercise 6.2, reaching any term of a geometric progression directly, however far out it sits. For the fixed-amount contrast this exercise builds on throughout, revisit Exercise 6.1 or the chapter introduction.