Solved Number Series with Shortcuts

Example

7, 12, 17, 22, ? The difference is +5 each time, so the answer is 27. Using nth term = a + (n − 1) x d gives 7 + 4 x 5 = 27.

Example

5, 8, 14, 23, 35, ? The differences are 3, 6, 9, 12, which are multiples of 3. The next difference is 15, so the answer is 35 + 15 = 50. Repeating the last difference of 12 gives 47, which is the built-in trap.

Example

3, 7, 15, 31, 63, ? Each term is the previous one doubled plus 1, so the answer is 127. Shortcut: every term is one less than a power of 2, and the next power is 128.

Example

0, 7, 26, 63, 124, ? Each term is n cubed minus 1 for n = 1 to 6, so the answer is 216 − 1 = 215. Shortcut: each term sits just below a cube, since 7 is near 8, 26 near 27 and 63 near 64.

Example

1, 2, 6, 21, 88, ? Here 1 x 1 + 1 = 2, 2 x 2 + 2 = 6, 6 x 3 + 3 = 21, 21 x 4 + 4 = 88, so the next is 88 x 5 + 5 = 445. Forgetting the +5 gives 440, which is the trap option.

Example

4, 30, 8, 26, 12, 22, 16, ? Odd positions run 4, 8, 12, 16 rising by 4. Even positions run 30, 26, 22 falling by 4. The blank is at an even position, so the answer is 22 − 4 = 18.

Example

1, 1, 2, 4, 7, 13, 24, ? Each term is the sum of the previous three, so the next is 7 + 13 + 24 = 44. Adding only the previous two gives 37, the classic trap.

Example

4, 9, 25, 49, 121, ? These are the squares of 2, 3, 5, 7 and 11, so the next is 13 squared = 169. The series skips 16 and 36 because 4 and 6 are not prime.

Example

2, 6, 12, 20, 30, 44, 56. The rule is n x (n + 1), giving 2, 6, 12, 20, 30, 42, 56. The sixth term should be 42, so 44 is wrong. The differences 4, 6, 8, 10, 14, 12 confirm it, since the two abnormal values sit on either side of 44.