Fill in the blanks with the numbers that complete the sequence, and

briefly explain the rule you used

2,3,5,5,8,7,11 _,_,_,17,_,_,_

3,4,3,4,5,3,4,5,_,_,_,_,_,7

1,2,1,1,2,3,2,1,1,2,3,4,3,_,_,1,2,_,_,_,_,_,2

Hints:

1. 2,3,5,5,8,7,11 _,_,_,17,_,_,_

Looks to me a sandwiched arithmetic sequences of 2,5,8,11,14,17... and 3,5,7,9,...

2. 3,4, 3,4,5, 3,4,5,_, _,_,_,_,7

3. 1,2,1, 1,2,3,2,1, 1,2,3,4,3,_,_, 1,2,_,_,_,_,_,2

890;891 897

The first sequence is: 2, 3, 5, 5, 8, 7, 11, _, _, _, 17, _, _, _.

To determine the missing numbers in this sequence, let's analyze the pattern:

- The first number is 2.
- The second number is 3, which is 1 more than the previous number.
- The third number is 5, which is 2 more than the previous number.
- The fourth number is 5 again since it repeats.
- The fifth number is 8, which is 3 more than the previous number.
- The sixth number is 7, which is 1 less than the previous number.
- The seventh number is 11, which is 4 more than the previous number.

Based on this pattern, we can see that the next number should be 11 + 5, which equals 16. Hence, the sequence should be:

2, 3, 5, 5, 8, 7, 11, 16, 21, 26, 17, 31, 36, _

The second sequence is: 3, 4, 3, 4, 5, 3, 4, 5, _, _, _, _, _, 7.

To identify the pattern, let's analyze the sequence:

- The first number is 3.
- The second number is 4, which is 1 more than the previous number.
- The third number is 3, which is 1 less than the previous number.
- The fourth number is 4 again since it repeats.
- The fifth number is 5, which is 1 more than the previous number.
- The sixth number is 3, which is 2 less than the previous number.
- The seventh number is 4 again.
- The eighth number is 5 again.

Based on this pattern, we can deduce that the numbers repeat in the pattern 3, 4, 3, 4, 5, with each repetition having the same numbers. Hence, the missing numbers should follow the same pattern:

3, 4, 3, 4, 5, 3, 4, 5, 6, 7, 8, 9, 10, 7.

The third sequence is: 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, _, _, 1, 2, _, _, _, _, _, 2.

To find the rule, let's examine the sequence:

- The first number is 1.
- The second number is 2, which is 1 more than the previous number.
- The third number is 1, which is 1 less than the previous number.
- The fourth number is 1 again.
- The fifth number is 2, which is 1 more than the previous number.
- The sixth number is 3, which is 1 more than the previous number.
- The seventh number is 2, which is 1 less than the previous number.
- The eighth number is 1 again.
- The ninth number is 1 again.
- The tenth number is 2.
- The eleventh number is 3.
- The twelfth number is 4, which is 1 more than the previous number.
- The thirteenth number is 3 again.

Based on this pattern, we can observe that each number appears for a certain number
of times and then switches to the next number. The sequence is as follows:
1 (1 time), 2 (1 time), 1 (2 times), 1 (2 times), 2 (2 times), 3 (1 time), 2 (1 time), 1 (2 times), 1 (2 times), 2 (2 times), 3 (3 times), 4 (1 time), 3 (3 times), 2 (1 time).

Hence, the missing numbers in the sequence should follow the same pattern:

1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 2.