Identifying the Pattern in the Sequence

The sequence 1, 1, 3, 2, 8, 5, 21, 13, ___, ___ does not follow a simple arithmetic or geometric progression. The numbers fluctuate, which suggests it is composed of two or more independent series interleaved together.

Separating the Interleaved Series

Let's split the main sequence into two sub-series based on their positions (odd and even):

• Series 1 (Odd Positions): 1, 3, 8, 21, ...
• Series 2 (Even Positions): 1, 2, 5, 13, ...

Analyzing the Pattern

Both series appear to follow the same recursive formula: $X_n = 3 \times X_{n-1} - X_{n-2}$. This means each new term is three times the previous term minus the term before that.

Checking Series 1:

• (3 × 3) - 1 = 9 - 1 = 8
• (3 × 8) - 3 = 24 - 3 = 21

Checking Series 2:

• (3 × 2) - 1 = 6 - 1 = 5
• (3 × 5) - 2 = 15 - 2 = 13

The pattern holds true for both series.

Calculating the Next Two Numbers

We apply the formula to find the next term for each series, which will be the two missing numbers in the main sequence.

1. Next term for Series 1: (3 × 21) - 8 = 63 - 8 = 55
2. Next term for Series 2: (3 × 13) - 5 = 39 - 5 = 34

Conclusion

The two missing numbers in the sequence are the next term from Series 1 followed by the next term from Series 2.

Therefore, the pair of numbers is 55, 34.