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Series and sums

The 40th term of 3, 7, 11, …; 1 + 2 + … + 100; the first 50 odd numbers; a geometric sum — each derived from pairing ends or shifting terms.

  • Quantitative reasoning
  • Untimed
  • Endless levels

How it plays

Each question is a sequence: find a term far along it, count its terms, or add them up. The sums of 1 to n, of the odd and even numbers, of squares and cubes, and of geometric series come back again and again.

New kinds unlock as you climb — geometric terms and sums, squares and cubes, a series that never ends — and n grows without end.

The wrong options are the real slips: n steps where there are n − 1, n² ÷ 2 for the triangle number, the last term taken for the sum, the halving forgotten. The working shows the derivation — pair the ends, or shift the series by one and subtract.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes the series and sums it shows quicker to derive. It claims nothing wider.

Learn

How to get better at series and sums: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Decide what kind of sequence it is. Subtract neighbours: the same difference each time is arithmetic. Divide neighbours: the same ratio each time is geometric. Then read what is asked — a term, a count, or a sum.
  2. Count steps, not terms. From term 1 to term n there are n − 1 steps; from the first term to the last there are (last − first) ÷ step of them.
  3. For a sum, reach for a derivation, not a memory: pair the ends for an arithmetic series, multiply by r and subtract for a geometric one.
  4. Check the size before you commit. A sum of n terms lies between n times the smallest and n times the largest; a geometric sum is about r ÷ (r − 1) times its last term — twice it for r = 2, one and a half times for r = 3.

Techniques and tricks

  • nth term: first plus n − 1 steps

    Term n of an arithmetic sequence is the first term plus (n − 1) times the step. The −1 is the whole trick: term 1 has had no steps.

    e.g. 3, 7, 11, …: term 40 is 3 + 39 × 4 = 159

  • Count the terms: steps plus one

    Divide the gap from first to last by the step to get the steps, then add one for the terms. Fence posts, not fence panels.

    e.g. 5, 12, 19, …, 145: (145 − 5) ÷ 7 = 20 steps, so 21 terms

  • Pair the ends

    An arithmetic series has the same average as the average of its first and last terms. The sum is that average times the number of terms. For 1 to n, it is n(n + 1) ÷ 2.

    e.g. 1 + … + 100 = 100 × 101 ÷ 2 = 5,050; 4, 7, …, 61 (20 terms): 20 × 65 ÷ 2 = 650

  • Odd numbers make squares, evens make n(n + 1)

    The first n odd numbers add to n²; each is the next L-shaped layer round a square. The first n even numbers are twice 1 to n, so n(n + 1). Find n first: the last odd number is 2n − 1, the last even 2n.

    e.g. 1 + 3 + … + 99: n = 50, so 2,500; 2 + 4 + … + 60: 30 × 31 = 930

  • Geometric sum: shift and subtract

    Multiply the sum by r: it is the same terms moved one along. Subtract, and everything cancels except the ends. S = (r × last − first) ÷ (r − 1). For r = 2 that is just twice the last minus the first.

    e.g. 3, 6, …, 384: 2 × 384 − 3 = 765; 4, 12, …, 972: (3 × 972 − 4) ÷ 2 = 1,456

  • Squares and cubes

    1² + … + n² = n(n + 1)(2n + 1) ÷ 6; cancel the 6 before multiplying. The cubes are prettier: 1³ + … + n³ is the square of 1 + … + n.

    e.g. 1² + … + 10² = 10 × 11 × 21 ÷ 6 = 385; 1³ + … + 10³ = 55² = 3,025

  • A series that never ends: S = a ÷ (1 − r)

    If each term is 1/q of the one before, the tail after the first term is the whole series shrunk by 1/q. So S = a + S/q, which gives S = a × q ÷ (q − 1).

    e.g. 18, 6, 2, …: S = 18 × 3 ÷ 2 = 27

As the levels climb

  • The kinds unlock in this order: the nth term, 1 + … + n and the odd numbers; then counting terms, the even numbers and arithmetic sums; then geometric terms and sums; then squares, cubes and a series that never ends. Four in ten questions come from the newest kind.
  • n grows as you climb: sequences run further, sums get longer, and from size 4 on the label some arithmetic sequences count down instead of up. The method does not change — a negative step is still added n − 1 times.
  • Geometric sequences stay short — a handful of terms, under a trillion — while the ratio creeps up from 2 and 3 as the size grows. Use powers you know — 2¹⁰ = 1,024, 3⁵ = 243 — rather than multiplying term by term.
  • Each step of size adds 8% to the time, up to twelve steps. Typed answers must be exact.

Traps to avoid

  • n steps where there are n − 1: 3 + 40 × 4 = 163 for the 40th term of 3, 7, 11. Term 1 has had no steps added.
  • n² ÷ 2 for the triangle number, or n(n − 1) ÷ 2: 1 + … + 100 is 5,050, not 5,000 or 4,950. The pairs add to n + 1, not n.
  • The last term taken for the sum, in a geometric series: 384 is the last term of 3, 6, …, 384; the sum is 765. And the halving forgotten for an arithmetic one gives twice the answer.
  • The odd numbers counted to the wrong n: 1 + 3 + … + 99 is 50², not 99². Find how many odd numbers there are — (99 + 1) ÷ 2 — before you square.
  • Squares and cubes swapped: n(n + 1)(2n + 1) ÷ 6 for cubes, or (n(n + 1) ÷ 2)² for squares. Check with n = 2: squares give 5, cubes give 9.

How to practise

  • In “Kinds of question”, drill “The nth term” and “How many terms” together; they are the same n − 1 seen from both ends.
  • Switch “Answer by” to “Typing the answer”. Answers must be exact, so you have to finish the arithmetic, not pick the one of the right size.
  • Raise “Options shown” towards 8 to put the off-by-one, the unhalved and the last-term answers beside the true one.
  • Read the working after each miss: it derives the formula each time, by pairing the ends or shifting the series, and that is the part worth keeping.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More quantitative reasoning

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