16,49,104,181

I always first check the sequence of differences to see
if I can recognize a pattern:

2nd term - 1st term =   49-16 = 33 = 11×3
3rd term - 2nd term =  104-49 = 55 = 11×5 
4th term - 3rd term = 181-104 = 77 = 11×7

So we can now see the pattern.

 1.    Start with 16, then
 2.    16 + 11×3  =   49

and the multiple of 11, which is 3, is 1 less than twice the term
number, which is 2. 

 3.    49 + 11×5  =  104 =

16 + 11×3 + 11×5

and the multiple of 11, which is 5, is 1 less than twice the term
number, which is 3.


 4.   104 + 11×7  =  181 =

16 + 11×3 + 11×5 + 11×7

and the multiple of 11, which is 7, is 1 less than twice the term
number, which is 4

Therefore to find the 20th term, we start with 16 and add
19 terms.  So the 20th term is

16+ =

Factor out 11 from the sun:

16+ =

Break up the sum into two sums:

16+ =

Factor out 2 from the first sum:

16+ =

The sum  is the sum of an arithmetic sequence
with a1 = 2 and a19 = 20, and d=1

Sn = (a1 + an)
S19 = (2 + 20) = (22) = 209

And  is the sum of 19 ones, which is 19.

Substituting in:

16+ =

We have

16+11·(2·209-19) =

4405.

Edwin