Algebra

We can set variables up to solve this problem. Let's set 

From the problem, we can write the equations

$S_5 = 5a_1 + (d + 2d + 3d + 4d) = 5a_1 + 10d\\S_{10} = 10a_1 + (d + 2d + \cdots + 9d) = 10a_1 + 45d\\\\\S_{15} = 15a_1 + (d + 2d + \cdots + 14d) = 15a_1 + 105d$

Simplifying the first 2 equations, we get the conditions

$\begin{cases} 5a_1 + 10d = \dfrac15\\ 10a_1 + 45d = \dfrac1{10} \end{cases}$

Solving this gives that d is -3/250 and a_1 is 8/125. 

Thus, we have

$S_{15} = 15a_1 + 105d = -\dfrac3{10}$

So -3/10 is our answer. 

Thanks! :)