The first five numbers in Patterns A and B are given. Pattern A: \( 5,7,10,14,19, \ldots \) Pattern B: 4, \( 8,12,16,20, \ldots \) What is the first number that is in both Pattern A and Pattern B? Enter the number in the box.

To find the first number that is in both Pattern A and Pattern B, we first need to identify the rules governing each pattern. **Pattern A:** The numbers are \( 5, 7, 10, 14, 19, \ldots \). Let's analyze the differences between consecutive terms: - \( 7 - 5 = 2 \) - \( 10 - 7 = 3 \) - \( 14 - 10 = 4 \) - \( 19 - 14 = 5 \) The differences are increasing by 1 each time. We can express the \( n \)-th term of Pattern A as follows: - \( a_1 = 5 \) - \( a_2 = 5 + 2 = 7 \) - \( a_3 = 7 + 3 = 10 \) - \( a_4 = 10 + 4 = 14 \) - \( a_5 = 14 + 5 = 19 \) The general formula for the \( n \)-th term can be derived, but for now, we can continue to generate terms until we find a match. **Pattern B:** The numbers are \( 4, 8, 12, 16, 20, \ldots \). This is an arithmetic sequence where each term increases by 4: - \( b_n = 4n \) Now, let's list the terms of both patterns until we find a common number. **Terms of Pattern A:** 1. \( 5 \) 2. \( 7 \) 3. \( 10 \) 4. \( 14 \) 5. \( 19 \) 6. \( 25 \) (next term: \( 19 + 6 = 25 \)) 7. \( 32 \) (next term: \( 25 + 7 = 32 \)) 8. \( 40 \) (next term: \( 32 + 8 = 40 \)) **Terms of Pattern B:** 1. \( 4 \) 2. \( 8 \) 3. \( 12 \) 4. \( 16 \) 5. \( 20 \) 6. \( 24 \) 7. \( 28 \) 8. \( 32 \) (next term: \( 4 \times 8 = 32 \)) Now we can see that the first common number in both patterns is \( 32 \). Thus, the first number that is in both Pattern A and Pattern B is \( \boxed{32} \).