1. Let's analyze the given series: 7, 20, 58, 171, ___. 2. We look for a pattern or rule that connects the terms. 3. Check the ratio between consecutive terms: - $\frac{20}{7} \approx 2.857$, - $\frac{58}{20} = 2.9$, - $\frac{171}{58} \approx 2.948$. 4. The ratios are increasing but not by a fixed amount, so let's check differences: - $20 - 7 = 13$, - $58 - 20 = 38$, - $171 - 58 = 113$. 5. Now, look at the ratios of these differences: - $\frac{38}{13} \approx 2.923$, - $\frac{113}{38} \approx 2.974$. 6. This suggests the differences multiply roughly by 3 each time. 7. Let's suppose the next difference is about $113 \times 3 = 339$. 8. Therefore, the next term: $171 + 339 = 510$. 9. Verify by approximate ratio: $\frac{510}{171} \approx 2.98$, aligns with previous trend. Final answer: The next number in the series is $\boxed{510}$.