Key takeaways
- A geometric sequence multiplies by the same number at every step, so the diagnostic move is dividing consecutive terms rather than subtracting them.
- The nth term is a × r to the power n − 1, and the exponent counts multiplication steps in exactly the way the n − 1 in an arithmetic rule counts addition steps.
- A negative common ratio makes the signs alternate, which is a giveaway that people routinely misread as two interleaved sequences.
- A ratio between −1 and 1 makes terms shrink, and an infinite run of shrinking terms can add up to a finite total.
- A difference table never settles on a geometric sequence, so if your differences keep growing in proportion, switch tests immediately rather than differencing again.
A geometric sequence multiplies by the same number at every step. That number is the common ratio, usually written r, and the test for it is division rather than subtraction. The University of Waterloo's CEMC sequences and series toolkit gives the general term as a × r raised to the power n − 1, and the finite sum as a(1 − rⁿ)/(1 − r), with a separate infinite form once |r| is below 1.
This is the second family in the number sequence puzzle catalogue, and it is worth separating from the constant-difference family for one practical reason: the diagnostic move is different, and using the wrong one wastes the time you were trying to save.
Divide, do not subtract
Take 6, 18, 54, 162. The differences are 12, 36, 108 — growing, and not obviously helpful. The ratios are 18 ÷ 6 = 3, 54 ÷ 18 = 3, 162 ÷ 54 = 3. Constant, so the sequence is geometric with a = 6 and r = 3.
There is a useful warning sign here. Differencing a geometric sequence produces another geometric sequence with the same ratio, so a difference table on this family never settles no matter how many rows you take. If your first differences are themselves growing in proportion rather than by a fixed amount, stop differencing and start dividing.
The simplest member of the family is doubling. The powers of 2 — 1, 2, 4, 8, 16, 32, 64, 128 — are catalogued as A000079 in the On-Line Encyclopedia of Integer Sequences, where the entry notes that these numbers appear on Old Babylonian cuneiform tablets, which makes them among the oldest recorded mathematical objects of any kind.
The nth term: counting multiplications
The rule is built the same way the arithmetic one is, by counting steps.
- Term 1 is a. No multiplications yet.
- Term 2 is a × r. One multiplication.
- Term n is a × rⁿ⁻¹. n − 1 multiplications.
For 6, 18, 54, 162 the rule is 6 × 3ⁿ⁻¹. The seventh term is 6 × 3⁶ = 6 × 729 = 4,374. Check the rule at n = 1 before using it: 6 × 3⁰ = 6, correct.
The sum of the first six terms, from the formula, is 6(3⁶ − 1)/(3 − 1) = 6 × 728 ÷ 2 = 2,184. Adding the terms directly gives 6 + 18 + 54 + 162 + 486 + 1,458, which totals 2,184 as well. Verifying a formula against a direct addition once, on a short run, is worth the thirty seconds; it catches an off-by-one in the exponent that would otherwise survive into every later answer.
Negative ratios alternate
A negative r flips the sign at every step. Take 5, −10, 20, −40: the ratio is −2 throughout, because −10 ÷ 5, 20 ÷ −10 and −40 ÷ 20 all give −2.
The sixth term is 5 × (−2)⁵ = 5 × (−32) = −160. Odd exponents keep the negative, even exponents cancel it, so the signs run positive, negative, positive, negative from the first term onwards.
This is the family's most commonly misread shape. A sequence that swings above and below zero looks like two interleaved strands, and solvers split it before testing for a single negative ratio. Run the division test first — it is one step and it resolves the question outright.
Ratios below one shrink, and the shrinking can be added up
When r sits between −1 and 1, terms get smaller. Take 80, 20, 5, 1.25, where r = ¼. The terms never reach zero, but they lose three-quarters of their size at every step, and the running total approaches a ceiling: 80 ÷ (1 − 0.25) = 80 ÷ 0.75 = 320/3, or about 106.67.
The oldest argument in this territory is Zeno's dichotomy paradox. MacTutor's biography of Zeno of Elea sets out the version in which reaching a destination requires first reaching the midpoint, then the midpoint of what remains, and so on without end. The distances covered are ½, ¼, ⅛ and onwards — a geometric sequence with a = ½ and r = ½, whose infinite sum is 0.5 ÷ 0.5 = 1, exactly the whole journey. MacTutor is careful to note that the mathematical sum does not dispose of the philosophical puzzle, which is still discussed; what it does show is that infinitely many positive terms can total a finite amount.
The everyday version: percentages are ratios
Any quantity changing by a fixed percentage per period is geometric. A 20% rise is a ratio of 1.2; a 20% fall is a ratio of 0.8. This is why a 20% rise followed by a 20% fall does not return you to where you started — 100 becomes 120, then 96 — a point developed in detail in percentages in your head.
It is also why long-run growth questions cannot be answered by adding. Something growing 7% a year has a ratio of 1.07, so after ten years it is multiplied by 1.07¹⁰, roughly 1.97 — close to double, and nothing like the 70% that adding would suggest. Recognising that a described change is a ratio rather than a difference is the single highest-value use of this family outside puzzles.
Practising the distinction
The skill worth building is not the formulas; it is the reflex of running both tests. Subtract first, and if the gaps are not constant, divide. If neither settles, the rule is probably recursive or interleaved, which is covered in the guide to finding a rule with difference tables.
IQ Test & Brain Training Games, our own app from Reign Creative, drills the arithmetic underneath this rather than the theory: Multiplication Rush and Division Dash for the ratio work, and the 75-second Number Pattern Challenge for rule-finding itself. Its 49-technique library includes a doubling-and-halving entry with a worked example and a named common mistake. Its listing also describes logic games built on patterns and rule discovery and an IQ-style test covering patterns and spatial reasoning, without saying whether any of them use figural, matrix or logic-grid items. One honest limit is worth stating: the listing describes the test's results as estimates based on app activities, not medical, psychological, clinical or professional diagnostic assessments, so a result there is a record of how you did on the app's own questions, not a measure of intelligence. The app page describes the app. More sequence material is gathered in the logic and pattern puzzles collection, and the wider subject sits in the education and brain training section.
Good to know
Frequently asked questions
How do you find the common ratio of a geometric sequence?
What does a negative common ratio do?
Why does an infinite geometric series have a finite sum?
Is compound growth a geometric sequence?
Sources
- Sequences and Series (opens in a new tab)Centre for Education in Mathematics and Computing, University of Waterloo — accessed
- A000079: Powers of 2, a(n) = 2^n (opens in a new tab)The On-Line Encyclopedia of Integer Sequences, OEIS Foundation Inc. — accessed
- Zeno of Elea (490 BC - 425 BC) - Biography (opens in a new tab)MacTutor History of Mathematics Archive, University of St Andrews — accessed
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