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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