SOLUTION: complete the series 4 9 19 34 54 ?

The first thing to do is to look for is the differences 
between the terms to see if you can see a pattern there:

2nd term minus 1st term =   9-4 =  5
3rd term minus 2nd term =  19-9 = 10
4th term minus 3rd term = 34-19 = 15
5th term minus 4th term = 54-34 = 20

We see a pattern
Turn all those subtractions into additions:
 4+ 5 =  9
 9+10 = 19
19+15 = 34
34+20 = 54

So the next one is

54+25 = 79

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This was a easy one.  Sometimes you can't see
a pattern with the differences, and you have to
try something else, maybe try the differences of
the differences, or the successive squares.
However always start by finding the differences
between successive terms.

Edwin