Write $v = \frac{2uw}{u+w}$ as $\frac{1}{v} = \frac{u+w}{2uw}$. This gives $\frac{1}{v} = \frac{1}{2} \left( \frac{1}{u} + \frac{1}{w} \right)$, or $\frac{1}{v} - \frac{1}{u} = \frac{1}{w} - \frac{1}{v}$. So look at the sequence of reciprocals of the given numbers: $\frac{1}{v} - \frac{1}{u} = \frac{1}{w} - \frac{1}{v}$ means that consecutive reciprocals differ by the same amount which we denote by $d$.

Hence the last reciprocal $\frac{1}{1/101} = 101$ can be obtained from the first one $\frac{1}{1/100} = 100$ by adding $d$ 20 times. Thus $101 = 100 + 20d$, yielding $d = \frac{1}{20}$. To obtain the 15th reciprocal we add $14d$ to the first one, 100, which gives $100 + 14 \cdot \frac{1}{20} = \frac{1007}{10}$. Therefore the 15th original number is $\frac{10}{1007}$.