1) Evaluate the following: \( \begin{array}{ll}\text { a) } \sum_{k=1}^{12} 5+2(k-3) & \text { e) } \sum_{k=1}^{12}-12+(r-2)(-2) \\ \text { b) } \sum_{k=5}^{12} 3(2)^{k-5} & \text { d) } \sum_{k=0}^{3} 2(3)(3)^{k}\end{array} \)

Let's evaluate each of the given summations step by step. ### Problem a) \[ \sum_{k=1}^{12} \left[5 + 2(k - 3)\right] \] **Step 1:** Simplify the expression inside the summation. \[ 5 + 2(k - 3) = 5 + 2k - 6 = 2k - 1 \] **Step 2:** Compute the sum. \[ \sum_{k=1}^{12} (2k - 1) = 2\sum_{k=1}^{12} k - \sum_{k=1}^{12} 1 \] - Sum of the first 12 natural numbers: \[ \sum_{k=1}^{12} k = \frac{12 \times 13}{2} = 78 \] - Sum of 1 added 12 times: \[ \sum_{k=1}^{12} 1 = 12 \] **Step 3:** Substitute the sums back. \[ 2 \times 78 - 12 = 156 - 12 = 144 \] **Answer:** **a) 144** --- ### Problem b) \[ \sum_{k=5}^{12} 3 \cdot 2^{k-5} \] **Step 1:** Change the index of summation for simplicity. Let \( m = k - 5 \), so when \( k = 5 \), \( m = 0 \), and when \( k = 12 \), \( m = 7 \). \[ \sum_{m=0}^{7} 3 \cdot 2^{m} = 3 \sum_{m=0}^{7} 2^{m} \] **Step 2:** Calculate the geometric series. \[ \sum_{m=0}^{7} 2^{m} = 2^{8} - 1 = 256 - 1 = 255 \] **Step 3:** Multiply by the constant. \[ 3 \times 255 = 765 \] **Answer:** **b) 765** --- ### Problem d) \[ \sum_{k=0}^{3} 2(3)(3)^{k} \] **Step 1:** Simplify the constant factors. \[ 2 \times 3 \times 3^{k} = 6 \times 3^{k} \] **Step 2:** Compute the sum. \[ 6 \sum_{k=0}^{3} 3^{k} = 6 \times (3^{0} + 3^{1} + 3^{2} + 3^{3}) = 6 \times (1 + 3 + 9 + 27) = 6 \times 40 = 240 \] **Answer:** **d) 240** --- ### Problem e) \[ \sum_{k=1}^{12} \left[-12 + (r - 2)(-2)\right] \] **Assumption:** It seems there's a typo with the variable \( r \). We'll assume it should be \( k \), making the expression: \[ \sum_{k=1}^{12} \left[-12 + (k - 2)(-2)\right] = \sum_{k=1}^{12} (-12 - 2k + 4) = \sum_{k=1}^{12} (-2k - 8) \] **Step 1:** Split the sum. \[ \sum_{k=1}^{12} (-2k) + \sum_{k=1}^{12} (-8) = -2 \sum_{k=1}^{12} k - 8 \times 12 \] **Step 2:** Calculate each part. \[ -2 \times 78 - 96 = -156 - 96 = -252 \] **Answer:** **e) -252**