1. Let's analyze the given series: 2, 5, 9, 19, 37, ___. 2. Find the differences between consecutive terms: $5 - 2 = 3$ $9 - 5 = 4$ $19 - 9 = 10$ $37 - 19 = 18$ 3. Differences are: 3, 4, 10, 18. Let's find differences of these differences: $4 - 3 = 1$ $10 - 4 = 6$ $18 - 10 = 8$ 4. Second level differences are 1, 6, 8, which do not show a clear pattern. Let's check the ratio or other relationships. 5. Look for a pattern: From 2 to 5: $2 imes 2 + 1 = 5$ From 5 to 9: $5 imes 1 + 4 = 9$ From 9 to 19: $9 imes 2 + 1 = 19$ From 19 to 37: $19 imes 1 + 18 = 37$ These mixed steps make no consistent pattern. 6. Alternatively, look for a pattern in multiplication alternating with addition: 2 to 5: $2 imes 2 + 1 = 5$ 5 to 9: $5 imes 1 + 4 = 9$ 9 to 19: $9 imes 2 + 1 = 19$ 19 to 37: $19 imes 1 + 18 = 37$ This is inconsistent. 7. Instead, notice the pattern in differences: 3, 4, 10, 18 If the next difference follows the addition pattern, check the differences between these differences: 4 - 3 = 1 10 - 4 = 6 18 - 10 = 8 If next difference increase is sum of 1,6,8 = 15, then next difference could be 18 + 15 = 33. 8. Therefore, next term = last term + next difference = $37 + 33 = 70$. 9. The next number in the series is **70**.