1. **Stating the problem:** We want to find a formula for the number of matches $T_n$ used to make $n$ hexagons joined side-by-side, where each additional hexagon shares one match with the previous one. 2. **Understanding the pattern:** - Figure 1 (1 hexagon) uses 6 matches. - Figure 2 (2 hexagons) uses 10 matches. - Figure 3 (3 hexagons) uses 14 matches. 3. **Observing the pattern:** Each new hexagon after the first adds 4 matches because it shares one match with the previous hexagon. 4. **Writing the formula:** The first hexagon uses 6 matches. Each additional hexagon adds 4 matches. So for $n$ hexagons: $$ T_n = 6 + 4(n - 1) $$ 5. **Simplifying the formula:** $$ T_n = 6 + 4n - 4 = 4n + 2 $$ 6. **Final formula:** $$ \boxed{T_n = 4n + 2} $$ This formula gives the total number of matches needed to make $n$ hexagons joined side-by-side with shared matches.