Each term in a geometric sequence equals the previous term multiplied by a constant common ratio r, so the nth term is a₁ times r raised to the power n minus one — that's the formula aₙ = a₁rⁿ⁻¹ that powers the Geometric Sequence Quiz. A number pattern quiz built around this rule shows you a first term, a common ratio, and a five-position row with one hidden slot, and your task is to compute the missing value with that formula. The browser-based Geometric Sequence Quiz tests exactly five disclosed cards that cover the realistic edge cases — a positive ratio, a negative ratio that flips the sign each step, a zero ratio that collapses the sequence, a unit ratio that holds the value steady, and a larger positive ratio. Each correct fill advances one card and awards 200 points, so entering the five literal answers produces exactly 1,000 points. The quiz accepts only plain signed numbers, treats empty or malformed entries as no-ops, and ends the run on a second distinct wrong value — the kind of strict numeric discipline that makes it useful practice rather than a casual click-through.
What a Geometric Sequence Quiz Actually Tests
Generic number pattern quizzes usually mix arithmetic jumps, alternating signs, fibonacci-like sums, and ad-hoc rules. A geometric sequence quiz narrows the field to the single recurrence aₙ = r · a_(n-1), which is the algebraic identity behind compound interest, repeated dilution, halving, and binary doubling. That focus matters because the rule has a clean explicit formula: aₙ = a₁ · r^(n-1). Once you recognize the form, the missing term on any card can be computed without guesswork, which is why the Geometric Sequence Quiz displays both the first term a₁ and the common ratio r on every card rather than hiding them behind a multiple-choice menu.
The quiz deliberately picks five ratios that exercise the corners of the formula. A ratio of 2 grows the sequence quickly, a ratio of -2 alternates sign at every step, a ratio of 0 collapses everything to zero after the first term, a ratio of 1 freezes every term at the first value, and a ratio of 3 amplifies by a larger factor. None of these edge cases is exotic in a classroom sense — they are the same fixtures that textbooks use to test whether a student really understands the rule or has memorized a particular example.
The Five Disclosed Cards and Their Missing Terms
Every card in the Geometric Sequence Quiz shows a first term, a common ratio, and five positions in order with one cell blanked out. The five cards are fixed fixtures, so the missing term on each one is the same every run. The table below restates the published cards and their answers so you can verify the rule before you start, and so you can see at a glance which card tests which edge case.
| Card | First term a₁ | Common ratio r | Hidden position | Missing term aₙ |
|---|---|---|---|---|
| 1 | 3 | 2 | 4th term | 24 |
| 2 | 5 | -2 | 4th term | -40 |
| 3 | 7 | 0 | 4th term | 0 |
| 4 | -4 | 1 | 4th term | -4 |
| 5 | 2 | 3 | 5th term | 162 |
The fifth term on card 5 sits one row past the others because the final card also doubles as the cap of the run. Submitting the five values 24, -40, 0, -4, and 162 in order clears every card and lands on exactly 1,000 points — the score is the sum of five literal 200-point awards, not a relative or scaled total that depends on time or attempt count.
How to Solve the Missing Term on Each Card
- Read the displayed first term a₁ and common ratio r on the current card, then decide whether to multiply step by step or jump straight to the explicit formula aₙ = a₁ · r^(n-1).
- Identify the highlighted missing position on the card. Cards 1–4 hide the fourth term (n = 4); card 5 hides the fifth term (n = 5).
- Type a signed integer or decimal into the input field. The parser accepts plain numbers such as -40 and 24 but rejects scientific notation, blanks, whitespace, and nonfinite values.
- Press Enter or click Check term to submit. Empty or malformed input is ignored, so it cannot spend an error or change the score.
- Repeat for all five cards. Once you submit 24, -40, 0, -4, and 162 in order, the score freezes at 1,000 points and the run is complete.
To verify card 1 by hand: aₙ = 3 · 2^(4-1) = 3 · 2^3 = 3 · 8 = 24. Card 2 follows the same template but flips sign at every step: aₙ = 5 · (-2)^3 = 5 · (-8) = -40. Card 3 collapses because any term past a₁ is a₁ · 0^(n-1) = 0 for n ≥ 2, so the fourth term is 0. Card 4 leaves every term equal to a₁ when r = 1, so -4 reappears at the missing position. Card 5 closes the run with aₙ = 2 · 3^(5-1) = 2 · 81 = 162.
Why Negative, Zero, and Unit Ratios Matter
A geometric sequence quiz that only used positive integer ratios would test the multiplication step but skip the sign and constant branches. The Geometric Sequence Quiz mixes r = 2, r = -2, r = 0, r = 1, and r = 3 so you have to think about what happens to the sign when r is negative, what happens to the value when r reaches zero, and what happens when the sequence becomes constant. Negative ratios in particular are easy to mis-handle because every step flips the sign and an off-by-one exponent produces a wrong sign at the missing position, which is one of the most common pattern-quiz errors in any source.
Zero and unit ratios are the two short-circuit cases. With r = 0, the sequence terminates at zero for any position past the first; with r = 1, the sequence never changes at all. Both rules are simple to apply, but both will trip you up if you try to compute them by stepping through every term rather than recognizing that the formula collapses the pattern. The quiz exposes these edge cases by design — they are the same edge cases the cited mathematics references flag when they introduce the explicit formula.
Score Logic, Errors, and Restart Controls
Each correct answer is worth exactly 200 points, and the cap is fixed at 1,000 points across the five cards. The first distinct numeric mistake is recoverable: the error count rises by one and the same card stays active for another attempt. A second distinct numeric mistake ends the run immediately, the input freezes at a deadlock state, and later submissions are blocked. Equivalent but typographically different entries such as 999 and 999.0 are deduplicated against the stored error signature, so they cannot count as a second mistake.
Blank input, whitespace-only input, exponential notation, and nonfinite values are atomic no-ops. They neither advance the round, change the score, nor spend an error. That means the only thing that can end your run is two distinct wrong numbers, so a single typo or accidental keystroke gives you a second chance rather than a fail. Press R with focus outside the editing field to restart: the score resets to zero, the input clears, and the first card reappears. The shared Boss Key from the surrounding game shell remains available throughout the run.
Formulas and Sources Behind the Quiz
The mathematical definition and formulas used by the quiz are cross-checked against OpenStax College Algebra 2e — Chapter 9 Key Equations and Mathematics LibreTexts Advanced Algebra — 9.3 Geometric Sequences and Series. Both references define a geometric sequence by the same recurrence aₙ = r · a_(n-1) and the same explicit form aₙ = a₁ · r^(n-1). The quiz cites them only for the mathematics; the scoring, error rules, and game fixtures are independent Lizely design choices.
Two support vectors are worth knowing if you want to double-check the sequence generator. A positive first term with a positive ratio (for example a₁ = 3, r = 2) yields strictly increasing positive values, while a positive first term with a negative ratio (for example a₁ = 5, r = -2) alternates sign at every step. The generator accepts only finite first terms and ratios with an integer length between one and twenty, normalizes negative zero for clean display, and rejects nonfinite intermediate values so it never returns a misleading result. Independent tests of eight literal evaluation vectors — positive, negative, fractional, zero, and unit ratios plus negative first terms — confirm that the production generator matches the formula on the page.