Seeing a method applied is the fastest way to learn it. Here are nine original series, from easy to harder, each solved the way you would solve it on paper. Cover the solution, try the series, then compare.
Example 1: steady steps
17, 23, 29, 35, 41, ?
The gaps are 6, 6, 6, 6. A constant gap means a fixed step. The next term is 41 + 6 = 47. Check: 17 + 6 = 23, and so on to 41, all consistent.
Example 2: growing gaps
1, 2, 4, 7, 11, 16, ?
The gaps are 1, 2, 3, 4, 5. They climb by 1, so the next gap is 6. The next term is 16 + 6 = 22. The tempting wrong answer is 21, which repeats the last gap (5) instead of continuing the growth.
Example 3: a constant ratio
2, 6, 18, 54, 162, ?
Each term is three times the last: 6 / 2 = 3, 18 / 6 = 3, 54 / 18 = 3, 162 / 54 = 3. The next term is 162 × 3 = 486.
Example 4: cubes
1, 8, 27, 64, 125, ?
These are 1³, 2³, 3³, 4³ and 5³. The next is 6³ = 216. The gaps (7, 19, 37, 61) and their gaps (12, 18, 24) are no use here; only the cube list helps, which is why memorising it is worth the effort.
Example 5: square-number gaps
3, 4, 8, 17, 33, 58, ?
The gaps are 1, 4, 9, 16, 25: the squares of 1 to 5. The next gap is 6² = 36, so the term is 58 + 36 = 94.
Example 6: sum of the three before
1, 2, 4, 7, 13, 24, ?
The gaps (1, 2, 3, 6, 11) are not regular. Try summing earlier terms: 1 + 2 + 4 = 7, 2 + 4 + 7 = 13, 4 + 7 + 13 = 24. The rule holds throughout, so the next term is 7 + 13 + 24 = 44.
Example 7: a pair of operations
5, 10, 8, 16, 14, 28, ?
The row goes up, down, up, down, so check an alternating rule. 5 doubles to 10, then takes away 2 to give 8, doubles to 16, takes away 2 to give 14, doubles to 28. The next step is take away 2, giving 26.
Example 8: two series woven together
3, 50, 6, 45, 9, 40, 12, 35, ?
The odd positions are 3, 6, 9, 12 (add 3). The even positions are 50, 45, 40, 35 (take away 5). Eight numbers are shown, so the ninth is an odd position: 12 + 3 = 15.
Example 9: two series with the same rule
2, 3, 4, 6, 8, 12, 16, ?
The odd positions are 2, 4, 8, 16 and the even positions are 3, 6, 12. Both double. Seven numbers are shown, so the next is the eighth, an even position: 12 × 2 = 24.
What to take from the nine
Notice how few moves there are. Gaps solved examples 1, 2 and 5; ratio solved 3; a memorised list solved 4; summing earlier terms solved 6; trying a pair of operations solved 7; splitting the row solved 8 and 9. The routine in the finding-the-rule article is those moves, tried in a fixed order.
For a scored set of series of this kind, take the JobCannon Number Series test. More practice by type sits on the number series practice page. All nine answers here were checked by program: the stated rule reproduces every shown number and yields the answer given.