We need the value of any term in the sequence with it's nth place in the sequence as well as the number of terms in the sequence.
Statement 1 ; The least term in the sequence is 8.
There is no Sufficient information on the number of terms the sequence has, thus statement 1 is NOT SUFFICIENT.

Statement 2; The average of (arithmetic mean) terms is 16.
The Average / arithmetic mean doesn't tell about how many term the sequence has, hence statement 2 is NOT SUFFICIENT.
Combining the two statements,
Average of sequence of consecutive terms = Median of the sequence = Average of the least and highest term = $$\frac{\left(8\ +\ highest\ terms\right)}{2}\ =\ 16$$
8 + highest term = (16 * 2)
Highest term = (16 * 2) - 8 ) = 32 - 8
Highest term = 24
The series includes all integers from 8 to 24 a total of 17 terms.
Sum of the series = 16 * 17 = 272

The two statement together are SUFFICIENT.
Option C is the correct answer.