Note that there are alternate ways to express the sequences besides those listed here.

1. There are two alternating rules for this sequence.
The first rule is to always add 1 to the previous number.
The second rule begins by adding 0 to the previous number; the amount added to the previous number is then incremented by 1 for each application of the rule. Therefore, the second rule will add 0, then add 1, then add 2, etc.

Solution:
0 ⁺¹ 1 ⁺⁰ 1 ⁺¹ 2 ⁺¹ 3 ⁺¹ 4 ⁺² 6 ⁺¹ 7 ⁺³ 10 ⁺¹ 11 ⁺⁴ 15

2. The rotating rule here is to subtract 6 to generate the first number, subtract 12 to generate the second, and add 10 to generate the third. The rule then repeats.

Solution:
36 ^-6 30 ^-12 18 ⁺¹⁰ 28 ^-6 22 ^-12 10 ⁺¹⁰ 20 ^-6 14

3. This are two interleaved sequences:
The first sequence is the squares, starting with 1².
The second sequence is 100 - the cubes, starting with 100 - 1³.

Solution:
1, 99, 4, 92, 9, 73, 16, 36

4. There are two alternating rules for this sequence.
The first rule begins by adding 1 to the previous number; the amount added to the previous number is then incremented by 1 for each application of the rule. Therefore, the first rule will add 1, then add 2, then add 3, etc.
The second rule begins by adding 7 to the previous number; the amount added to the previous number is then decremented by 1 for each application of the rule. Therefore, the second rule will add 7, then add 6, then add 5, etc.

Solution:
0 ⁺¹ 1 ⁺⁷ 8 ⁺² 10 ⁺⁶ 16 ⁺³ 19 ⁺⁵ 24 ⁺⁴ 28 ⁺⁴ 32 ⁺⁵ 37

5. There are two alternating rules for this sequence.
The first rule is to always add 3 to the previous number.
The second rule is to always add 5 to the previous number.

Solution:
1 ⁺³ 4 ⁺⁵ 9 ⁺³ 12 ⁺⁵ 17 ⁺³ 20 ⁺⁵ 25 ⁺³ 28 ⁺⁵ 33

6. There are two alternating rules for this sequence.
The first rule begins by adding 1 to the previous number; the amount added to the previous number is then incremented by 2 for each application of the rule. Therefore, the first rule will add 1, then add 3, then add 5, etc.
The second rule begins by subtracting 1 from the previous number; the amount subtracted from the previous number is then incremented by 1 for each application of the rule. Therefore, the second rule will subtract 1, subtract 2, subtract 3, etc.

Solution:
15 ⁺¹ 16 ^-1 15 ⁺³ 18 ^-2 16 ⁺⁵ 21 ^-3 18 ⁺⁷ 25 ^-4 21 ⁺⁹ 30

7. There is a single rule for this sequence. The rule is to take the sum of the previous two numbers; if the sum is 100 or higher, then the sum is reduced by 100.
For example, we start with 57, 32. The sum of those is 89, which is unchanged because it is under 100. Next is 57, 32, 89. Summing 32+89 gives 121 which, because it is over 100, is reduced to 21.

Solution:
57, 32, 89, 21, 10, 31, 41, 72, 13

8. There is a single rule for this sequence. The rule is to take difference of the previous two numbers; if the difference is under 0, then the number is increased by 100.
For example, we start with 27, 57. The difference is -30. Since it is under 0, 100 is added to give 70

Solution:
27, 57, 70, 87, 83, 4, 79, 25

9. There are two alternating rules for this sequence.
The first rule is to take the ones digit from the previous number.
The second rule is to multiply the previous number by 7.

Solution:
38 ^·8 8 ^x7 56 ^·6 6 ^x7 42 ^·2 2 ^x7 14 ^·4 4 ^x7 28

10. There is a single rule for this sequence. The rule changes the previous amount by 1, then 2, then 3, then 4. Whether the change is an addition or subtraction changes as well--it rotates add, add, subtract:

Solution:
11 ⁺¹ 12 ⁺² 14 ^-3 11 ⁺⁴ 15 ⁺⁵ 20 ^-6 14 ⁺⁷ 21 ⁺⁸ 29 ^-9 20