How can the sequence be deciphered?

The sum of the first 10 terms of an arithmetic series is 185.

The sum of its fourth and eighth terms is two times the sixth term.

2a + 55d = 185

Assuming d to be 3

2a = 185 - 175

2a = 10

a = 5

5, 8, 11, 14, ...

an = 3n + 2

The second sentence is of no help in solving this problem.

Let n be the first term and d be the difference.

The first ten terms amount to n + n+d + n+2d + n+3d +...n+9d

10n + 55d = 185

The fourth term is n+3d

The 8th term is n+7d

The 6th term is n+5d

So from the statement of the problem n+3d + n+7d = 2(n+5d)

Simplifying this 2n +10d = 2n +10d Notice that this is true for all n and all d. In other words, all arithmetic series will have the sum of the 4th and 8th terms equal to twice the 6th term. This then gives us no extra information.

Back to the earlier part:

10n + 55d = 185

Lets just try a few example values for n and for d

n = 14, d = 1

14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 = 185 17+21 = 38 = 19*2

n = 5, d = 3

5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32 = 185 14 + 26 = 40 = 20*2

So, there is not a single possible answer to the question.