Question
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
Ask by Munoz Allan. in South Africa
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( S_{100} = 10200 \) and the first three terms are 3, 8, and 15.
Solution
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: \( 3, 8, 15 \)
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Simplify this solution Bonus Knowledge
To find \( S_{100} \), we can plug \( n = 100 \) into the formula \( S_n = n^2 + 2n \): \[ S_{100} = 100^2 + 2(100) = 10000 + 200 = 10200 \] So, \( S_{100} = 10200 \). For the first three terms of the sequence, we can calculate \( S_1 \), \( S_2 \), and \( S_3 \): \[ S_1 = 1^2 + 2(1) = 1 + 2 = 3 \] \[ S_2 = 2^2 + 2(2) = 4 + 4 = 8 \] \[ S_3 = 3^2 + 2(3) = 9 + 6 = 15 \] Thus, the first three terms are \( 3, 8, 15 \).
