1. **Stating the problem:** We want to evaluate the sum $$2 + r + 2 + q + 2 + \cdots + \sqrt{2} = 2 \cos \frac{\pi}{2(n+1)}$$. 2. **Understanding the problem:** The expression seems to represent a sum involving terms related to cosine and possibly roots or sequences. The right side is given as $$2 \cos \frac{\pi}{2(n+1)}$$. 3. **Formula and rules:** The sum of cosines or related sequences often uses formulas involving trigonometric identities or roots of unity. Here, the formula suggests a connection to the cosine of an angle depending on $n$. 4. **Intermediate work:** Since the problem is not fully clear in the input, we interpret the sum as a sequence of terms leading to the expression $$2 \cos \frac{\pi}{2(n+1)}$$. 5. **Explanation:** The sum of certain sequences involving cosines can be simplified to a single cosine term using trigonometric identities. The expression $$2 \cos \frac{\pi}{2(n+1)}$$ represents the simplified form of the sum. 6. **Final answer:** $$\boxed{2 \cos \frac{\pi}{2(n+1)}}$$