(e) Consider the number pattern: \( 4 \times 7 ; 7 \times 15 ; 10 \times 23 ; 13 \times 31 ; \ldots \ldots \) (1) Determine the \( n \)th term. (2) Determine the 50 th term

To analyze the given number pattern \( 4 \times 7 ; 7 \times 15 ; 10 \times 23 ; 13 \times 31 ; \ldots \), we can break it down into two sequences: the first factors and the second factors. ### Step 1: Identify the first factors The first factors are: - \( 4, 7, 10, 13, \ldots \) This sequence can be observed as an arithmetic sequence where: - The first term \( a_1 = 4 \) - The common difference \( d = 3 \) The \( n \)th term of an arithmetic sequence can be expressed as: \[ a_n = a_1 + (n-1)d \] Substituting the values: \[ a_n = 4 + (n-1) \cdot 3 = 4 + 3n - 3 = 3n + 1 \] ### Step 2: Identify the second factors The second factors are: - \( 7, 15, 23, 31, \ldots \) This sequence can also be observed as an arithmetic sequence where: - The first term \( b_1 = 7 \) - The common difference \( d = 8 \) The \( n \)th term of this arithmetic sequence can be expressed as: \[ b_n = b_1 + (n-1)d \] Substituting the values: \[ b_n = 7 + (n-1) \cdot 8 = 7 + 8n - 8 = 8n - 1 \] ### Step 3: Combine the sequences Now, we can express the \( n \)th term of the entire pattern as: \[ T_n = a_n \times b_n = (3n + 1)(8n - 1) \] ### Step 4: Expand the expression Expanding \( T_n \): \[ T_n = (3n + 1)(8n - 1) = 3n \cdot 8n + 3n \cdot (-1) + 1 \cdot 8n + 1 \cdot (-1) \] \[ = 24n^2 - 3n + 8n - 1 = 24n^2 + 5n - 1 \] Thus, the \( n \)th term is: \[ T_n = 24n^2 + 5n - 1 \] ### Step 5: Determine the 50th term To find the 50th term, substitute \( n = 50 \) into the formula: \[ T_{50} = 24(50^2) + 5(50) - 1 \] Calculating \( 50^2 \): \[ 50^2 = 2500 \] Now substituting: \[ T_{50} = 24(2500) + 5(50) - 1 \] Calculating \( 24 \times 2500 \): \[ 24 \times 2500 = 60000 \] Calculating \( 5 \times 50 \): \[ 5 \times 50 = 250 \] Now putting it all together: \[ T_{50} = 60000 + 250 - 1 = 60249 \] ### Final Answers (1) The \( n \)th term is: \[ T_n = 24n^2 + 5n - 1 \] (2) The 50th term is: \[ T_{50} = 60249 \]