1. **Stating the problem:** We have two sequences describing the number of chairs around tables arranged side by side. The first sequence claims the number of chairs is $2n+1$ for $n$ tables, but the data shows this is incorrect. The second sequence claims the number of chairs is $4n+1$ for $n$ tables, but this is also incorrect. 2. **Goal:** Find the correct formulas for the number of chairs surrounding $n$ tables arranged side by side for each sequence. 3. **Analyzing the first sequence:** - Given data points: - $n=1$, chairs = 8 - $n=2$, chairs = 10 - $n=3$, chairs = 12 - $n=4$, chairs = 14 - $n=82$, chairs = 170 4. **Check if the pattern is linear:** - Differences between chairs: - $10 - 8 = 2$ - $12 - 10 = 2$ - $14 - 12 = 2$ This constant difference suggests a linear formula: $$\text{chairs} = an + b$$ 5. **Find $a$ and $b$ using two points:** - Using $n=1$, chairs = 8: $$a(1) + b = 8$$ - Using $n=2$, chairs = 10: $$a(2) + b = 10$$ Subtract first from second: $$2a + b - (a + b) = 10 - 8 \Rightarrow a = 2$$ Plug back to find $b$: $$2(1) + b = 8 \Rightarrow b = 6$$ 6. **Correct formula for first sequence:** $$\boxed{\text{chairs} = 2n + 6}$$ 7. **Verify with $n=82$:** $$2(82) + 6 = 164 + 6 = 170$$ which matches the data. 8. **Analyzing the second sequence:** - Given data points: - $n=1$, chairs = 12 - $n=2$, chairs = 16 - $n=3$, chairs = 20 - $n=4$, chairs = 24 - $n=76$, chairs = 312 9. **Check if the pattern is linear:** - Differences between chairs: - $16 - 12 = 4$ - $20 - 16 = 4$ - $24 - 20 = 4$ Again, a linear formula: $$\text{chairs} = an + b$$ 10. **Find $a$ and $b$ using two points:** - Using $n=1$, chairs = 12: $$a(1) + b = 12$$ - Using $n=2$, chairs = 16: $$a(2) + b = 16$$ Subtract first from second: $$2a + b - (a + b) = 16 - 12 \Rightarrow a = 4$$ Plug back to find $b$: $$4(1) + b = 12 \Rightarrow b = 8$$ 11. **Correct formula for second sequence:** $$\boxed{\text{chairs} = 4n + 8}$$ 12. **Verify with $n=76$:** $$4(76) + 8 = 304 + 8 = 312$$ which matches the data. **Final answers:** - First sequence: $$\text{chairs} = 2n + 6$$ - Second sequence: $$\text{chairs} = 4n + 8$$