SOLUTION: Trying to find the patterns of; 1, 1, 9, 9, ? 1, 9, 9, 81, ? 6, 0, 0, 0, ? 5, 5, 25, 25, ? 1, 2, 20, 40, ?

1, 1, 9, 9,

Two 1's then two 9's, then 2 1's, then two 9's, over and over, etc.
That is:

1, 1, 9, 9, 1, 1, 9, 9, 1, 1, 9, 9, 1, 1, 9, 9,...

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1, 9, 9, 81, 

The first two terms are 1 and 9.

After that, each term is the product of the two preceding terms:

1, 9, 1×9=9, 9×9=81, 9×81=729, 81×729=59049, 729×59049=43046721, etc. 

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6, 0, 0, 0, ?

6, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, etc.

6, then three zeros, then 6, then three zeros, then 6, then three zeros, etc. 

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5, 5, 25, 25, ?

The first two terms are 5's.  After that each pair of terms are both the next higher power of 5. 

51, 51, 52, 52, 53, 53, 54, 54, 55, 55, etc. 

 or

5, 5, 25, 25, 125, 125, 625, 625, 3125, 3125, etc.


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1, 2, 20, 40, ?

The first two terms are 1 and 2.

After that, the next two terms, respectively, are 20 times the preceding 
two terms, respectively.
  
1, 2, 20, 40, 400, 800, 8000, 16000, etc.

Edwin