Number Series Questions and Answers

Answer: D

The missing number is 65.

Each term is one more than a cube:

\[ \begin{aligned} 2 &= 1^{3} + 1,\quad 9 = 2^{3} + 1,\quad 28 = 3^{3} + 1 \\ 65 &= 4^{3} + 1,\quad 126 = 5^{3} + 1 \end{aligned} \]

so the gap is filled by \(4^{3} + 1 = 65\). 63 is \(4^{3} - 1\), and 57 and 53 do not fit the pattern; the differences 7, 19, 37, 61 also confirm 65, since they rise by 12, 18, 24.

Answer: C

The missing number is 20.

Each term is the product of two consecutive numbers:

\[ \begin{aligned} &1 \times 2,\ 2 \times 3,\ 3 \times 4, \\ &4 \times 5,\ 5 \times 6 \end{aligned} \]

which gives 2, 6, 12, 20, 30. Equivalently, the differences grow by 2 each time: 4, 6, 8, 10, so the gap after 12 must be 8 and the gap before 30 must be 10. Each wrong option breaks this: 15 would leave gaps of 3 and 15, 18 gaps of 6 and 12, and 25 gaps of 13 and 5.

Answer: B

The wrong number is 43.

The differences between consecutive terms should rise by 2 each time: 9 - 2 = 7, 18 - 9 = 9, 29 - 18 = 11, so the next difference is 13 and the next term should be 29 + 13 = 42. From 42 the pattern continues correctly: 42 + 15 = 57 and 57 + 17 = 74.

With 43 in the series the differences become 14 and 14, breaking the pattern. 9, 29 and 74 all fit the corrected series 2, 9, 18, 29, 42, 57, 74, so they are not wrong.

Answer: B

The missing number is 6.

The differences between terms are 2, 4, 6, 8, 10, rising by 2 each time:

\[ \begin{aligned} 0 + 2 &= 2 \\ 2 + 4 &= 6 \\ 6 + 6 &= 12 \\ 12 + 8 &= 20 \\ 20 + 10 &= 30 \end{aligned} \]

Equivalently, the terms are n(n - 1) for n = 1 to 6: 0, 2, 6, 12, 20, 30. 4, 8 or 10 in the gap would make the differences uneven (2, 2, 8 or 2, 6, 4 or 2, 8, 2), breaking the steady rise of 2 that the rest of the series shows.

Answer: B

The next number is 97.

71, 73, 79, 83 and 89 are consecutive prime numbers. The next prime after 89 is 97: 90, 92, 94, 96 and 98 are even, 91 = 7 × 13, 93 = 3 × 31 and 95 = 5 × 19.

The other options are not prime: 96 is even, 98 is even, and 99 = 9 × 11. So only 97 continues the series.

A quick test for primes below 100: a number is prime if it is not divisible by 2, 3, 5 or 7, the primes up to the square root of 100. 97 passes all four tests.

Answer: A

The missing number is 43.

The differences alternate between +9 and +2:

12 (+9) 21 (+2) 23 (+9) 32 (+2) 34 (+9) 43

Another way to see it: the pairs 12 and 21, 23 and 32, 34 and 43 are each a number and its digits reversed. 36 or 37 would add 2 or 3 again, and 42 does not continue either pattern.

Checking the rule on every step confirms it: 21 - 12 = 9, 23 - 21 = 2, 32 - 23 = 9, 34 - 32 = 2, so the next step is +9 again.

Answer: C

The wrong number is 50.

The series is the squares of consecutive odd numbers:

\[ 1^{2}, 3^{2}, 5^{2}, 7^{2}, 9^{2} = 1, 9, 25, 49, 81 \]

So 50 should be 49. The other terms, 1, 25 and 81, all fit the pattern, so none of them is the wrong number.

Answer: D

The next number is 37.

The terms are prime numbers, taking every second prime: the primes run 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37. After 29 the next prime is 31, which is skipped, and the one after that is 37.

31 is the trap, the very next prime, but the series skips one prime each time. 35 and 39 are not prime at all.

Answer: A

The next number is 19.

The differences between consecutive terms alternate between +3 and +2:

1 to 4 is +3, 4 to 6 is +2, 6 to 9 is +3, 9 to 11 is +2, 11 to 14 is +3 and 14 to 16 is +2.

The step after +2 is +3, so 16 + 3 = 19. Taking 16 + 2 = 18 repeats the last step instead of alternating, and 17 or 16 do not follow the pattern at all.

Answer: C

The missing number is 742.

The differences alternate between cubes added and squares subtracted, each base falling by one:

\[ \begin{aligned} 24 + 8^{3} &= 536,\ 536 - 7^{2} = 487 \\ 487 + 6^{3} &= 703,\ 703 - 5^{2} = 678 \\ 678 + 4^{3} &= 742 \end{aligned} \]

736 and 836 would need differences of 58 and 158, and 842 a difference of 164; none is the cube of 4 (64), which the pattern requires next.