Yes—the Geometric Sequence Quiz includes negative, zero, and unit common ratios: its five fixed cards use r = −2, 0, 1, 2, and 3. That makes the answer yes for both special cases in your question, and the cases appear in actual play rather than merely in the instructions. Every card shows the first term, the common ratio, five positions, and one highlighted missing term, so each answer can be checked directly from the page. The route is fixed: enter 24, −40, 0, −4, and 162 in that order. The negative-ratio card begins at 5 and uses −2; the zero-ratio card begins at 7 but collapses to zero; and the unit-ratio card repeats a first term of −4. Use the Geometric Sequence Quiz for a transparent browser exercise based on the standard geometric-sequence rule. A correct answer advances one fixed card and earns 200 points, so all five cards produce exactly 1,000 points. The quiz also has a firm mistake rule: the first distinct wrong numeric value creates one recoverable error, while a second distinct numeric error closes the run. You can therefore practice signs, zero-ratio behavior, constant sequences, and positive growth without wondering whether those ratios are actually present.
Which ratio cases are included?
The five questions are fixed game fixtures rather than random draws. On each card, read the first term and ratio separately: a negative first term does not mean the common ratio is negative. The quiz deliberately combines those cases so you can see how the same rule behaves when the multiplier is negative, zero, one, or positive. It displays five positions on every card and hides one of them, which leaves enough context to recompute the missing term rather than guess a pattern.
| Card | First term a₁ | Common ratio r | Fixed missing answer | Behavior demonstrated |
|---|---|---|---|---|
| 1 | 3 | 2 | 24 | Positive ratio; the fourth term is hidden. |
| 2 | 5 | −2 | −40 | Negative ratio; the signs alternate. |
| 3 | 7 | 0 | 0 | Zero ratio; later terms collapse to zero. |
| 4 | −4 | 1 | −4 | Unit ratio; the value remains constant. |
| 5 | 2 | 3 | 162 | Positive ratio; the sequence grows. |
This set directly covers your query. Card 2 supplies a negative common ratio, card 3 supplies a zero ratio, and card 4 supplies a unit ratio. Card 4 also shows why the first term and multiplier must be read independently: its first term is −4, but its ratio is 1. The two positive-ratio cards round out the set with straightforward multiplicative growth.
The formula behind each missing term
A geometric sequence repeatedly multiplies the preceding term by one constant, the common ratio r. Its recursive form is aₙ = r aₙ₋₁, and its explicit form is aₙ = a₁rⁿ⁻¹. These definitions are supported by OpenStax College Algebra 2e — Chapter 9 Key Equations and Mathematics LibreTexts Advanced Algebra — 9.3 Geometric Sequences and Series. Those references support the mathematical rule, not the product's scoring, interface, or game rules.
The negative-ratio card provides one worked check. Its first term is 5, its ratio is −2, and its missing term is the fourth position. Substituting into the explicit formula gives a₄ = a₁rⁿ⁻¹ = 5(−2)³ = 5(−8) = −40. The result stays negative because an odd power of a negative number is negative.
The other special cases can be recognized before calculation. With r = 0, every term after the first is zero because each new term is the previous term multiplied by zero. With r = 1, every term matches the first term. A negative ratio changes the sign with each multiplication, so sign placement is part of finding the answer rather than an optional formatting detail.
How to finish all five cards
- Open the game and identify the current card number, first term, common ratio, and highlighted hidden position. Keep the displayed formula available while you work.
- Find the hidden term by applying the multiplier for that position. You may multiply repeatedly or use aₙ = a₁rⁿ⁻¹. For negative answers, include the sign in your reasoning; for the zero-ratio card, remember that the first term can be nonzero while the hidden later term is zero.
- Type the exact value into the signed-number field. The fixed answers are integers, so enter plain numbers such as 24 or −40 rather than labels, extra symbols, or exponential notation.
- Press Enter or choose Check term. A correct value advances the current card and awards 200 points. Pointer and touch users can select the input and button, while keyboard players can tab into the field and press Enter.
- Submit the fixed answer sequence in order: 24, −40, 0, −4, and 162. The fifth correct answer completes all five fixed cards and produces exactly 1,000 points.
- If an answer is wrong, correct the same card before entering a second distinct wrong numeric value. The first error is recoverable, but a second distinct error closes the quiz. To restart, press R while focus is outside the editing field; this rebuilds the first card, clears the input and errors, and resets the score to zero.
How unusual entries and wrong answers are handled
Input handling separates harmless formatting from a genuine mathematical mistake. An empty or whitespace-only entry, nonnumeric text, exponential notation, a nonfinite value, or another malformed entry is ignored as an atomic no-op. It cannot advance the round, alter the score, or spend an error. You can therefore replace an ignored entry with a properly formatted number without losing a life.
A valid number that is not the missing term is treated differently: it creates one recoverable error. The stored error signature combines the round and the parsed numeric value. Repeating the same wrong number therefore does not lock the quiz, and numerically equivalent spellings such as 999 and 999.0 are deduplicated. A second distinct numeric error anywhere in the run closes the quiz. Once the run is complete or closed by an error, later submissions are frozen until you restart.
Controls, restart, and what the score measures
The responsive card keeps the current card number, first term, common ratio, formula, five sequence positions, score, error count, and feedback visible. That transparency lets you audit the answer instead of trusting a hidden result. The input field and Check term button support pointer and touch controls, while tabbing, typing, and pressing Enter provide the keyboard route. R restarts the quiz when focus is not in an editing field, and the shared Boss Key remains available through the existing game shell.
No account or upload is required. It is educational entertainment rather than a graded exam, placement test, tutoring guarantee, or aptitude assessment. The 1,000-point score measures completion of these five disclosed cards only; it does not place the player in a broader performance category.
What to remember before you play
The direct answer is therefore unambiguous: every run includes a negative ratio, a zero ratio, and a unit ratio because the cards are fixed. Before submitting, check three things—the multiplier, the hidden position, and the sign. If an entry is ignored as malformed, correct its format rather than assuming an error was charged. If a numeric answer is genuinely wrong, use the remaining recoverable error to enter the corrected term before a second distinct mistake ends the run.
Related reading: Grammar Errors vs Grammatical Errors: Quiz Guide.