Is it possible to insert exactly 125 Harmonic Means between $$-\frac{1}{49}$$ and $$\frac{1}{2009}$$? If so, what is the $$6^{\text{th}}$$ HM?

PLACEHOLDER ANSWER - the answer is "not possible". The reciprocals $$-49$$ and 2009 must be the ends of an AP with 125 terms between them, so the common difference is $$\frac{2009 + 49}{126} = \frac{49}{3}$$. The 3rd inserted term of this AP is $$-49 + 3 \times \frac{49}{3} = 0$$, whose reciprocal is undefined. So 125 harmonic means cannot be inserted. (Formally, the 6th term of the AP would be 49, giving an HM of $$\frac{1}{49}$$.)