1. **Problem statement:** For the matchstick patterns, find: i) The number of matchsticks in the nth pattern. ii) The number of matchsticks in the 50th pattern. 2. **Understanding the pattern:** The pattern shows a hexagonal chain growing by adding matchsticks in each step. 3. **Formula for the nth pattern:** The number of matchsticks in the nth pattern follows the formula: $$\text{Number of matchsticks} = 5n + 1$$ This is because each new pattern adds 5 matchsticks to the previous total, starting from 6 matchsticks at pattern 1. 4. **Calculate the 50th pattern:** Substitute $n=50$ into the formula: $$5 \times 50 + 1 = 250 + 1 = 251$$ 5. **Final answers:** i) Number of matchsticks in the nth pattern is $$5n + 1$$. ii) Number of matchsticks in the 50th pattern is $$251$$.