Number Series Questions and Answers

Answer: A

Correct answer: 18

The terms shrink by a constant factor rather than by a constant amount: 1458 over 486 is 3, and the same holds for every neighbouring pair, so each term is the previous one divided by 3. Dividing the last term, 54 over 3 is 18. As a check, multiplying the answer by 3 returns 54, the term before it. Dividing from the small end avoids the large numbers entirely and makes the constant factor obvious.

Answer: B

Correct answer: 54

The terms shrink by a constant factor rather than by a constant amount: 4374 over 1458 is 3, and the same holds for every neighbouring pair, so each term is the previous one divided by 3. Dividing the last term, 162 over 3 is 54. As a check, multiplying the answer by 3 returns 162, the term before it. The terms of a halving series must eventually reach a number that will not divide evenly, which limits how long such a series can run.

Answer: C

Correct answer: 37

The gaps are 3, 5, 7, 9, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is the square of its position plus one: position 1 gives 2 and position 5 gives 26. The next position is 6, which gives 37. As a check, the same rule applied to the earlier positions reproduces 2, 5, 10, 17, 26 exactly, so it accounts for every term shown and not merely the last. Perfect squares are worth recognising on sight up to about twenty, since they appear throughout this topic.

Answer: D

Correct answer: 48

The gaps are 5, 7, 9, 11, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is one less than the square of its position: position 1 gives 3 and position 5 gives 35. The next position is 6, which gives 48. As a check, the same rule applied to the earlier positions reproduces 3, 8, 15, 24, 35 exactly, so it accounts for every term shown and not merely the last. The differences between consecutive squares are the odd numbers, which is the quickest way to confirm a square series.

Answer: A

Correct answer: 42

The gaps are 4, 6, 8, 10, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is the square of its position plus the position itself: position 2 gives 2 and position 6 gives 30. The next position is 7, which gives 42. As a check, the same rule applied to the earlier positions reproduces 2, 6, 12, 20, 30 exactly, so it accounts for every term shown and not merely the last. A square series grows faster than any arithmetic one but more slowly than a geometric one, which places it at a glance.

Answer: B

Correct answer: 100

The gaps are 12, 20, 28, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is the square of an even number: position 1 gives 4 and position 4 gives 64. The next position is 5, which gives 100. As a check, the same rule applied to the earlier positions reproduces 4, 16, 36, 64 exactly, so it accounts for every term shown and not merely the last. Adding a constant to each square shifts the whole series without changing its differences, so the difference test still identifies it.

Answer: C

Correct answer: 81

The gaps are 8, 16, 24, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is the square of an odd number: position 1 gives 1 and position 4 gives 49. The next position is 5, which gives 81. As a check, the same rule applied to the earlier positions reproduces 1, 9, 25, 49 exactly, so it accounts for every term shown and not merely the last. Subtracting one from each square gives numbers that factorise neatly, which is sometimes the easier way to see the pattern.

Answer: D

Correct answer: 126

The gaps are 7, 19, 37, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is the cube of its position plus one: position 1 gives 2 and position 4 gives 65. The next position is 5, which gives 126. As a check, the same rule applied to the earlier positions reproduces 2, 9, 28, 65 exactly, so it accounts for every term shown and not merely the last. Cubes grow quickly and are worth knowing to about six, beyond which the series rarely runs.

Answer: A

Correct answer: 210

The gaps are 18, 36, 60, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is the cube of its position minus the position: position 2 gives 6 and position 5 gives 120. The next position is 6, which gives 210. As a check, the same rule applied to the earlier positions reproduces 6, 24, 60, 120 exactly, so it accounts for every term shown and not merely the last. The differences between consecutive cubes are themselves a second-order series, so a cube rule is hard to see from the differences alone.

Answer: B

Correct answer: 28

The gaps are 2, 3, 4, 5, 6, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is the sum of all the whole numbers up to its position: position 1 gives 1 and position 6 gives 21. The next position is 7, which gives 28. As a check, the same rule applied to the earlier positions reproduces 1, 3, 6, 10, 15, 21 exactly, so it accounts for every term shown and not merely the last. Triangular numbers count the dots in a growing triangle, and their differences are simply the counting numbers.