This is how I did it.  You may have some other method.

I started out dividing each term by the preceding term
to find out how much we're multiplying by each time to
get the next term.

7/2 = 3 1/2
26/7 = 3 5/7
101/26 = 3 23/26
400/101 = 3 97/101

Looking at those numbers shows that they're getting 
closer and closer to multiplying by 4.
So I asked myself, How much less is it each time
than multiplying by 4?  So I investigated this way:

2*4 = 8.  The next term, 7, is 1 less than 8.
7*4 = 28.  The next term, 26, is 2 less than 28.
26*4 = 104.  The next term, 101, is 3 less than 104.
101*4 = 404.  The next term, 400, is 4 less than 404. 

From that I got the pattern

2*4-1 = 8-1 = 7
7*4-2 = 28-2 = 26
26*4-3 = 104-3 = 101
101*4-4 = 404-4 = 400
400*4-5 = 1600-5 = 1595  

------------------------------

I noticed that the 1st, 3rd and 5th terms formed
the pattern 7,8,9

Then I noticed that 21 was 7*3 and 72 was 8*9
and that 3 is 3 to the 1 power and 9 is 3 to the 2 power.

7
7*31 = 7*3 = 21
8
8*32 = 8*8 = 72
9
9*33 = 9*27 = 243

This one is similar to the previous one.

I noticed that the 1st, 3rd and 5th terms formed
the pattern 7,8,9

Then I noticed that 16 was 2 more than multiplying
7 by 2.
Then I noticed that 27 was 3 more than multiplying
8 by 3
So I figured that the next term after 9 would be 4 more than
multiplying 9 by 4.  That would be 4 more than 36 or 40

7
7*2 = 14, 14+2 = 16
8
8*3 = 24, 24+3 = 27
9
9*4 = 36, 36+4 = 40

Edwin