1. **Stating the problem:** We have a sequence of patterns made from matchsticks forming equilateral triangles. The number of matchsticks for the first six patterns is given as 3, 7, 11, 15, 19, 23. 2. **Observing the pattern:** The number of matchsticks increases by 4 for each subsequent pattern. This means the sequence is arithmetic with a common difference $d=4$. 3. **Finding the number of matchsticks for pattern 10:** Since the first term $T_1=3$ and the difference $d=4$, the formula for the $n$th term of an arithmetic sequence is: $$T_n = T_1 + (n-1)d$$ Substituting $n=10$: $$T_{10} = 3 + (10-1) \times 4 = 3 + 9 \times 4 = 3 + 36 = 39$$ So, pattern 10 requires 39 matchsticks. 4. **Finding the general formula for $T_n$:** Using the arithmetic sequence formula: $$T_n = 3 + (n-1) \times 4 = 3 + 4n - 4 = 4n - 1$$ Therefore, the formula for the number of matchsticks in pattern $n$ is: $$\boxed{T_n = 4n - 1}$$ This formula allows you to find the number of matchsticks for any pattern number $n$ in the sequence.