1. **Stating the problem:** We are given sequences of tables and chairs and need to identify and correct the incorrect formulas for the number of chairs based on the number of tables. 2. **Top row sequence:** - Given: Number of tables $n$ and chairs for $n=1,2,3,4$ are $6,8,10,12$ respectively. - Pattern: Chairs increase by 2 for each additional table. - Formula guess: $2n + 4$ (since for $n=1$, $2(1)+4=6$). - Check for $n=83$: $2(83)+4=170$ which matches the given number. - **Conclusion:** The formula for chairs is $2n + 4$. 3. **Middle row sequence:** - Given: Number of tables $n$ and chairs for $n=1,2,3,4$ are $6,10,14,18$ respectively. - Pattern: Chairs increase by 4 for each additional table. - Given formula: $4n + 1$. - Check for $n=1$: $4(1)+1=5$ but given is 6, so formula is incorrect. - Check for $n=2$: $4(2)+1=9$ but given is 10, so formula is incorrect. 4. **Find correct formula for middle row:** - Let formula be $an + b$. - Using $n=1$, chairs=6: $a(1)+b=6$. - Using $n=2$, chairs=10: $a(2)+b=10$. - Subtract equations: $2a + b - (a + b) = 10 - 6$ gives $a=4$. - Substitute $a=4$ into first equation: $4 + b=6$ so $b=2$. - Correct formula: $$\text{Chairs} = 4n + 2$$ - Check for $n=123$: $4(123)+2=494$ matches given. 5. **Bottom row sequence:** - Given: Number of tables $n$ and chairs for $n=1,2,3$ are $3,6,9$ respectively. - Pattern: Chairs increase by 3 for each additional table. - Formula: $3n$. **Final answers:** - Top row formula: $$\text{Chairs} = 2n + 4$$ - Middle row corrected formula: $$\text{Chairs} = 4n + 2$$ - Bottom row formula: $$\text{Chairs} = 3n$$