Study the following linear number pattern : 87; 78; 69; ... 1 Determine the value of \( \mathrm{T}_{13} \). 2 Which term has a value of 15 ? 3 Which term is the last term smaller than -60 ? 4 Which term is the first term greater than -94 ? 5 Which term is the first negative term?

To solve the problem, we first need to identify the pattern in the given sequence: \( 87, 78, 69, \ldots \). 1. **Determine the common difference**: - The difference between consecutive terms is: - \( 78 - 87 = -9 \) - \( 69 - 78 = -9 \) - Thus, the common difference \( d = -9 \). 2. **General formula for the \( n \)-th term**: - The formula for the \( n \)-th term of an arithmetic sequence is given by: \[ T_n = a + (n - 1) \cdot d \] - Here, \( a = 87 \) (the first term) and \( d = -9 \). Now, we can use this formula to answer the questions. ### 1. Determine the value of \( T_{13} \): \[ T_{13} = 87 + (13 - 1) \cdot (-9) \] Calculating this: \[ T_{13} = 87 + 12 \cdot (-9) = 87 - 108 = -21 \] ### 2. Which term has a value of 15? We need to find \( n \) such that: \[ T_n = 15 \] Using the formula: \[ 15 = 87 + (n - 1) \cdot (-9) \] Rearranging gives: \[ 15 - 87 = (n - 1) \cdot (-9) \implies -72 = (n - 1) \cdot (-9) \] Dividing both sides by -9: \[ n - 1 = 8 \implies n = 9 \] ### 3. Which term is the last term smaller than -60? We need to find the largest \( n \) such that: \[ T_n < -60 \] Using the formula: \[ 87 + (n - 1) \cdot (-9) < -60 \] Rearranging gives: \[ (n - 1) \cdot (-9) < -60 - 87 \implies (n - 1) \cdot (-9) < -147 \] Dividing by -9 (and flipping the inequality): \[ n - 1 > 16.33 \implies n > 17.33 \] Thus, the largest integer \( n \) is 17. ### 4. Which term is the first term greater than -94? We need to find the smallest \( n \) such that: \[ T_n > -94 \] Using the formula: \[ 87 + (n - 1) \cdot (-9) > -94 \] Rearranging gives: \[ (n - 1) \cdot (-9) > -94 - 87 \implies (n - 1) \cdot (-9) > -181 \] Dividing by -9 (and flipping the inequality): \[ n - 1 < 20.11 \implies n < 21.11 \] Thus, the smallest integer \( n \) is 21. ### 5. Which term is the first negative term? We need to find the smallest \( n \) such that: \[ T_n < 0 \] Using the formula: \[ 87 + (n - 1) \cdot (-9) < 0 \] Rearranging gives: \[ (n - 1) \cdot (-9) < -87 \] Dividing by -9 (and flipping the inequality): \[ n - 1 > 9.67 \implies n > 10.67 \] Thus, the smallest integer \( n \) is 11. ### Summary of Results: 1. \( T_{13} = -21 \) 2. The term with a value of 15 is \( T_9 \). 3. The last term smaller than -60 is \( T_{17} \). 4. The first term greater than -94 is \( T_{21} \). 5. The first negative term is \( T_{11} \).