Advanced Number Series
The difference ladder, the ratio test, and the series that defeat both
Free trail · Analytical · ages 13–14 · 6 stations
The Difference Ladder
Stop guessing at series. Run the Difference Ladder instead: a fixed order of tests, cheapest first. You climb a rung only when the rung below fails.
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Write the gaps underneath the terms. If they are constant, the series is arithmetic and the nth term is a + (n−1)d. Two matching gaps are not proof; check every gap you are given.
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Gaps not constant? Take the gaps of the gaps. A constant second difference means a quadratic rule of the form an² + bn + c, and that constant equals 2a. Constant third difference means a cubic.
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If the gaps are exploding rather than settling, divide each term by the one before it. A constant ratio means a geometric rule. If the ratio itself climbs 2, 3, 4, 5 you are looking at a factorial-style series.
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Test the term against n itself: n², n³, n² ± n, n³ − 1, n! and so on. 7, 26, 63, 124 looks wild until you notice it is n³ − 1 for n = 2, 3, 4, 5.
This is one station of 6 in the trail, and finishing the trail earns an Explorer badge. Create a free account to play the rest →
Inside the app
The station above is the real thing — and it is one of 6. Inside the app your child plays the whole trail — every station is something they touch, drag and try, each one earns a checkmark, finishing the trail awards an Explorer badge, and the Practice Zone builds a fresh set of questions every visit. Progress is saved to their player, so they can stop and pick it up later.
What your child does, station by station
Learning mode has no timers, no scores and no wrong answers — every station is something the child touches and moves. This trail is one of the free ones, so it opens on any account.
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The Difference Ladder
Flip cards
Stop guessing at series. Run the Difference Ladder instead: a fixed order of tests, cheapest first. You climb a rung only when the rung below fails.
A named, ordered procedure that cracks the large majority of series. Flip every card.
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Watch a Quadratic Give Itself Away
Slider explorer
This series follows n² + n + 1. Slide through it and watch the first differences climb while the second difference refuses to move.
Slide through the terms and watch the second difference stay stubbornly constant.
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When the Ladder Fails
Try it
Each of these defeats the difference ladder. Climb to rung 3 or 4: divide neighbours, or test the term against its own position number.
Exploding gaps mean you are on the wrong rung. Switch to ratios, products or position formulas.
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Woven and Alternating Series
Try it
Rung 5. If the terms lurch up and down, stop treating it as one series. Split by position and solve the odd and even streams separately.
When a series swings wildly, suspect two calm series interleaved.
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Diagnose Before You Solve
Sorting baskets
Speed in a series question comes from diagnosis, not arithmetic. Look at how fast the terms grow and sort each one into the test that will crack it.
The real skill is choosing the right test in the first two seconds. Sort each series by its diagnosis.
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Practice Zone
Practice Zone
Ten questions on this topic — a proper practice set, fresh every time.
Ten questions on this topic. New ones every time you come back.
Free printable worksheet
Ten questions from this topic on one page, with an answer key at the bottom — a PDF you can print for homework, a car journey or a classroom. No account needed and nothing to sign up for. The Practice Zone in the app builds a fresh set every visit; this is a fixed set you can hold.
More Analytical for ages 13–14
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