QUESTION 1 1.1 Consider the arithmetic sequence: \( 8 ; 15 ; 22 ; \ldots \ldots \ldots \ldots \ldots \ldots \ldots \) 1.1.1 \( \quad \) Determine the \( 36^{\text {th }} \) term 1.1 .2 Calculate the sum of the first 36 terms. 1.1 .3 If it is given that \( \mathrm{T}_{72}+\mathrm{T}_{72-m}=786 \), determine the value of \( m \).

To solve the problem, we will analyze the arithmetic sequence given and use the properties of arithmetic sequences to find the required terms and sums. ### Known Conditions The arithmetic sequence provided is: - First term (\( a \)) = 8 - Common difference (\( d \)) = 15 - 8 = 7 ### 1.1.1 Determine the \( 36^{\text{th}} \) term The formula for the \( n^{\text{th}} \) term of an arithmetic sequence is given by: \[ T_n = a + (n - 1) \cdot d \] Substituting \( n = 36 \): \[ T_{36} = 8 + (36 - 1) \cdot 7 \] Calculating this step by step: \[ T_{36} = 8 + 35 \cdot 7 \] \[ T_{36} = 8 + 245 \] \[ T_{36} = 253 \] ### 1.1.2 Calculate the sum of the first 36 terms The formula for the sum of the first \( n \) terms of an arithmetic sequence is: \[ S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d) \] Substituting \( n = 36 \): \[ S_{36} = \frac{36}{2} \cdot (2 \cdot 8 + (36 - 1) \cdot 7) \] Calculating this step by step: \[ S_{36} = 18 \cdot (16 + 35 \cdot 7) \] Calculating \( 35 \cdot 7 \): \[ 35 \cdot 7 = 245 \] Now substituting back: \[ S_{36} = 18 \cdot (16 + 245) \] \[ S_{36} = 18 \cdot 261 \] Calculating the final sum: \[ S_{36} = 4698 \] ### 1.1.3 Determine the value of \( m \) given \( T_{72} + T_{72 - m} = 786 \) First, we need to find \( T_{72} \): \[ T_{72} = a + (72 - 1) \cdot d \] Substituting the values: \[ T_{72} = 8 + (72 - 1) \cdot 7 \] Calculating: \[ T_{72} = 8 + 71 \cdot 7 \] Calculating \( 71 \cdot 7 \): \[ 71 \cdot 7 = 497 \] Now substituting back: \[ T_{72} = 8 + 497 = 505 \] Now we can set up the equation: \[ T_{72} + T_{72 - m} = 786 \] Substituting \( T_{72} \): \[ 505 + T_{72 - m} = 786 \] Solving for \( T_{72 - m} \): \[ T_{72 - m} = 786 - 505 = 281 \] Now we need to find \( m \) such that: \[ T_{72 - m} = a + (72 - m - 1) \cdot d \] Substituting the known values: \[ 281 = 8 + (72 - m - 1) \cdot 7 \] Simplifying: \[ 281 = 8 + (71 - m) \cdot 7 \] Subtracting 8 from both sides: \[ 273 = (71 - m) \cdot 7 \] Dividing by 7: \[ 39 = 71 - m \] Solving for \( m \): \[ m = 71 - 39 = 32 \] ### Final Answers 1.1.1 \( T_{36} = 253 \) 1.1.2 \( S_{36} = 4698 \) 1.1.3 \( m = 32 \)