We can solve for S_15, the sum of the first 15 terms, using the properties of arithmetic series and the given information about S_5 and S_10.

Here's how to approach this problem:

Formula for Arithmetic Series Sum:

The sum (S_n) of an arithmetic series can be calculated using the formula:

S_n = n/2 * (a_1 + a_n)

where:

n = number of terms

a_1 = first term

a_n = nth term (last term in this case)

Relating S_5 and S_10:

We are given that:

S_5 = 1/5 (sum of the first 5 terms)

S_10 = 1/10 (sum of the first 10 terms)

Finding the Difference (S_10 - S_5):

Subtracting S_5 from S_10, we can eliminate the first term (a_1) from the equation:

S_10 - S_5 = (1/10) - (1/5)

This represents the sum of terms from the 6th term (a_6) to the 10th term (a_10) of the arithmetic sequence.

Simplifying the Difference:

(1/10) - (1/5) = (1 - 2) / 10 = -1/10

Understanding the Difference:

The difference (-1/10) represents the sum of the next 5 terms (a_6 to a_10) after the first 5 terms (a_1 to a_5). Since it's negative, it implies that the common difference (d) between terms in the sequence is negative.

Finding the Sum of Next 5 Terms (S_10 - S_5):

We can express the difference (-1/10) using the formula for the sum of an arithmetic series with n = 5 (number of terms from 6th to 10th):

-1/10 = 5/2 * (a_6 + a_{10})

Since we know the common difference (d) is negative, the sum of the next 5 terms (a_6 + a_{10}) is also negative.

Solving for S_15:

To find S_15 (sum of the first 15 terms), we can build upon the relationship between S_10 and S_5:

S_15 = S_10 + (Sum of terms from 11th to 15th)

We already know S_10 (1/10) and the relationship between S_10 and S_5 (difference representing the sum of terms from 6th to 10th).

The sum of terms from the 11th to 15th term will be similar to the sum of terms from the 6th to 10th term (both sets of 5 terms with a negative common difference).

Therefore, the sum of terms from the 11th to 15th term will also be -1/10.

Finding S_15:

S_15 = (1/10) + (-1/10) = 0

Therefore, the sum of the first 15 terms (S_15) is 0.