# Is $$|r| < 1$$?

Algebra Level 3

$\displaystyle \large \sum_{n \ = \ 2}^{\infty} \left(1 + c\right)^{- n} = 2$

What value of $$c$$ makes the above equation true? If the answer can be expressed as $$\dfrac{\sqrt{p} - q}{r}$$ for positive integers $$p$$, $$q$$, and $$r$$, and square-free $$p$$, find $$(p + 1)(q + 2)(r + 3)$$.

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