1. **State the problem:** We need to find the sum of the fractions $\frac{5}{6} + \frac{2}{7} + \frac{1}{2}$.\n\n2. **Formula and rules:** To add fractions, they must have a common denominator. The common denominator is the least common multiple (LCM) of the denominators. Here, denominators are 6, 7, and 2.\n\n3. **Find the LCM:**\n- Prime factors: $6 = 2 \times 3$, $7$ is prime, $2$ is prime.\n- LCM is $2 \times 3 \times 7 = 42$.\n\n4. **Convert each fraction to have denominator 42:**\n$$\frac{5}{6} = \frac{5 \times 7}{6 \times 7} = \frac{35}{42}$$\n$$\frac{2}{7} = \frac{2 \times 6}{7 \times 6} = \frac{12}{42}$$\n$$\frac{1}{2} = \frac{1 \times 21}{2 \times 21} = \frac{21}{42}$$\n\n5. **Add the fractions:**\n$$\frac{35}{42} + \frac{12}{42} + \frac{21}{42} = \frac{35 + 12 + 21}{42} = \frac{68}{42}$$\n\n6. **Simplify the fraction:**\n- Find the greatest common divisor (GCD) of 68 and 42, which is 2.\n- Cancel common factor 2:\n$$\frac{\cancel{2} \times 34}{\cancel{2} \times 21} = \frac{34}{21}$$\n\n7. **Final answer:**\n$$\frac{5}{6} + \frac{2}{7} + \frac{1}{2} = \frac{34}{21}$$\nThis is an improper fraction and can also be written as $1 \frac{13}{21}$.