Given that \( S_{n}=n^{2}+2 n \) 2.1.1 Determine \( S_{100} \) 2.1.2 Determine the first three terms of the ori

To solve the problem, we will follow the steps outlined: ### Step 1: Extract Known Conditions We are given the formula for \( S_n \): \[ S_n = n^2 + 2n \] We need to determine \( S_{100} \) and the first three terms of the sequence. ### Step 2: Solve for \( S_{100} \) To find \( S_{100} \), we substitute \( n = 100 \) into the formula: \[ S_{100} = 100^2 + 2 \cdot 100 \] Calculating this step by step: 1. Calculate \( 100^2 \): \[ 100^2 = 10000 \] 2. Calculate \( 2 \cdot 100 \): \[ 2 \cdot 100 = 200 \] 3. Add the two results together: \[ S_{100} = 10000 + 200 = 10200 \] ### Step 3: Determine the First Three Terms of the Sequence The first three terms correspond to \( S_1 \), \( S_2 \), and \( S_3 \). 1. **Calculate \( S_1 \)**: \[ S_1 = 1^2 + 2 \cdot 1 = 1 + 2 = 3 \] 2. **Calculate \( S_2 \)**: \[ S_2 = 2^2 + 2 \cdot 2 = 4 + 4 = 8 \] 3. **Calculate \( S_3 \)**: \[ S_3 = 3^2 + 2 \cdot 3 = 9 + 6 = 15 \] ### Final Results - \( S_{100} = 10200 \) - The first three terms of the sequence are \( S_1 = 3 \), \( S_2 = 8 \), and \( S_3 = 15 \). Thus, the answers are: - \( S_{100} = 10200 \) - First three terms: \( 3, 8, 15 \)