Number Series Questions and Answers

Answer: C

Correct answer: 56

The gaps are 3, 6, 10, 15, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is the running total of the triangular numbers: position 1 gives 1 and position 5 gives 35. The next position is 6, which gives 56. As a check, the same rule applied to the earlier positions reproduces 1, 4, 10, 20, 35 exactly, so it accounts for every term shown and not merely the last. Each triangular number is the sum of all the whole numbers up to its position, which gives a direct formula.

Answer: D

Correct answer: 63

The gaps are 2, 4, 8, 16, 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 a power of two: position 1 gives 1 and position 5 gives 31. The next position is 6, which gives 63. As a check, the same rule applied to the earlier positions reproduces 1, 3, 7, 15, 31 exactly, so it accounts for every term shown and not merely the last. Powers of two are worth knowing well beyond the range of most series questions, since they underpin several families.

Answer: A

Correct answer: 65

The gaps are 2, 4, 8, 16, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is one more than a power of two: position 1 gives 3 and position 5 gives 33. The next position is 6, which gives 65. As a check, the same rule applied to the earlier positions reproduces 3, 5, 9, 17, 33 exactly, so it accounts for every term shown and not merely the last. Adding or subtracting one from a power of two leaves the doubling almost visible, which is how the family is spotted.

Answer: B

Correct answer: 243

The gaps are 6, 18, 54, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is three raised to the power of its position: position 1 gives 3 and position 4 gives 81. The next position is 5, which gives 243. As a check, the same rule applied to the earlier positions reproduces 3, 9, 27, 81 exactly, so it accounts for every term shown and not merely the last. A series whose terms are one less than a power of two has differences that are themselves powers of two.

Answer: C

Correct answer: 1024

The gaps are 12, 48, 192, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is four raised to the power of its position: position 1 gives 4 and position 4 gives 256. The next position is 5, which gives 1024. As a check, the same rule applied to the earlier positions reproduces 4, 16, 64, 256 exactly, so it accounts for every term shown and not merely the last. Powers of three grow faster than powers of two and are recognised by the same method of trial.

Answer: D

Correct answer: 720

The gaps are 1, 4, 18, 96, which follow no constant pattern, so the rule lies in the terms rather than in the steps between them. Each term is the product of every whole number up to its position: position 1 gives 1 and position 5 gives 120. The next position is 6, which gives 720. As a check, the same rule applied to the earlier positions reproduces 1, 2, 6, 24, 120 exactly, so it accounts for every term shown and not merely the last. A factorial series is identified by dividing each term by the one before and finding that the quotient rises by one each time.

Answer: A

Correct answer: 34

Each term is the sum of the two before it. Testing this in the middle of the series, 2 plus 3 is 5, and 3 plus 5 is 8. Applying it to the last pair, 13 plus 21 is 34. As a check, subtracting 21 from the answer leaves 13, which is the term two places back, exactly as the rule requires. Adding the two previous terms is a rule that leaves no trace in the differences, so it has to be tested directly on the terms themselves.

Answer: B

Correct answer: 29

Each term is the sum of the two before it. Testing this in the middle of the series, 1 plus 3 is 4, and 3 plus 4 is 7. Applying it to the last pair, 11 plus 18 is 29. As a check, subtracting 18 from the answer leaves 11, which is the term two places back, exactly as the rule requires. The Fibonacci rule can start from any pair, and recognising the structure matters more than recognising the particular numbers.

Answer: C

Correct answer: 47

Each term is the sum of the two before it. Testing this in the middle of the series, 3 plus 4 is 7, and 4 plus 7 is 11. Applying it to the last pair, 18 plus 29 is 47. As a check, subtracting 29 from the answer leaves 18, which is the term two places back, exactly as the rule requires. In a series of this kind the ratio between neighbouring terms drifts towards a fixed value, which is a useful way to spot the family at a glance.

Answer: D

Correct answer: 50

Each term is the sum of the two before it. Testing this in the middle of the series, 2 plus 5 is 7, and 5 plus 7 is 12. Applying it to the last pair, 19 plus 31 is 50. As a check, subtracting 31 from the answer leaves 19, which is the term two places back, exactly as the rule requires. Each term here depends on two predecessors rather than one, which is what distinguishes this family from every other in the topic.