Given the arithmetic series: \( 7+12+17+\ldots \) 2.1.1 Determine the value of \( T_{91} \) 2.1.2 Calculate \( S_{91} \) 2.1.3 Calculate the value of \( n \) for which \( T_{n}=517 \)

Let's analyze the given arithmetic series: \( 7, 12, 17, \ldots \). **Common Characteristics:** - **First term (\(a\))**: 7 - **Common difference (\(d\))**: 5 (since each term increases by 5) ### 2.1.1. Determine the value of \( T_{91} \) The \( n^{th} \) term of an arithmetic sequence is given by: \[ T_n = a + (n-1)d \] Plugging in the values: \[ T_{91} = 7 + (91 - 1) \times 5 = 7 + 90 \times 5 = 7 + 450 = 457 \] **Answer:** \( T_{91} = 457 \) --- ### 2.1.2. Calculate \( S_{91} \) The sum of the first \( n \) terms of an arithmetic sequence is given by: \[ S_n = \frac{n}{2} \times (a + T_n) \] From part 2.1.1, we know \( T_{91} = 457 \). Plugging in the values: \[ S_{91} = \frac{91}{2} \times (7 + 457) = \frac{91}{2} \times 464 = 91 \times 232 = 21,\!112 \] **Answer:** \( S_{91} = 21,\!112 \) --- ### 2.1.3. Calculate the value of \( n \) for which \( T_{n} = 517 \) Using the formula for the \( n^{th} \) term: \[ 517 = 7 + (n - 1) \times 5 \] \[ 517 - 7 = (n - 1) \times 5 \] \[ 510 = (n - 1) \times 5 \] \[ n - 1 = \frac{510}{5} = 102 \] \[ n = 103 \] **Answer:** \( n = 103 \)