Write down the next term in each of the following patterns: \( 3.1 .11 ; 2 ; 4 ; 2 ; 9 ; 2 ; 16 ; 2 ; 25 ; 2 ; \ldots \) \( 3.1 .2 \frac{2+1}{3^{2}-3} ; \frac{3+1}{4^{2}-3} ; \frac{4+1}{5^{2}-3} ; \frac{5+1}{6^{2}-3} ; \ldots \ldots \)

Let's analyze each pattern step by step to find the next term. ### Pattern 1: \( 3.1, 0.11, 2, 4, 2, 9, 2, 16, 2, 25, 2, \ldots \) 1. **Identify the sequence:** - The sequence alternates between two types of terms: one set of numbers and the number 2. - The first set of numbers is: \( 3.1, 0.11, 2, 4, 9, 16, 25 \). - The second set is consistently \( 2 \). 2. **Analyze the first set:** - The first set of numbers appears to be perfect squares: - \( 3.1 \) does not fit the pattern. - \( 0.11 \) does not fit the pattern. - \( 2 = 1^2 + 1 \) - \( 4 = 2^2 \) - \( 9 = 3^2 \) - \( 16 = 4^2 \) - \( 25 = 5^2 \) 3. **Next term in the first set:** - Following the pattern of perfect squares, the next term after \( 25 \) (which is \( 5^2 \)) would be \( 6^2 = 36 \). 4. **Next term in the overall sequence:** - Since the sequence alternates, the next term after \( 25 \) (which is followed by \( 2 \)) will be \( 36 \). ### Conclusion for Pattern 1: The next term is \( 36 \). --- ### Pattern 2: \( 3.1, 0.2, \frac{2+1}{3^{2}-3}, \frac{3+1}{4^{2}-3}, \frac{4+1}{5^{2}-3}, \frac{5+1}{6^{2}-3}, \ldots \) 1. **Identify the sequence:** - The terms are of the form \( \frac{n+1}{(n+2)^2 - 3} \) where \( n \) starts from \( 2 \). 2. **Analyze the terms:** - For \( n = 2 \): \( \frac{2+1}{3^2 - 3} = \frac{3}{6} = 0.5 \) - For \( n = 3 \): \( \frac{3+1}{4^2 - 3} = \frac{4}{13} \) - For \( n = 4 \): \( \frac{4+1}{5^2 - 3} = \frac{5}{22} \) - For \( n = 5 \): \( \frac{5+1}{6^2 - 3} = \frac{6}{33} \) 3. **Next term in the sequence:** - For \( n = 6 \): \[ \frac{6+1}{7^2 - 3} = \frac{7}{49 - 3} = \frac{7}{46} \] ### Conclusion for Pattern 2: The next term is \( \frac{7}{46} \). --- ### Final Answers: 1. The next term in the first pattern is \( 36 \). 2. The next term in the second pattern is \( \frac{7}{46} \).