Question
Use the formula \( S=n^{2} \) to find the sum of \( 1+3+5+\cdots+551 \). (Hint: To find \( n \), add 1 to the last term and divide by 2.) \( 1+3+5+\cdots+551=\square \) (Simplify your answer.)
Ask by Powell Vaughan. in Mexico
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The sum of the series \( 1 + 3 + 5 + \cdots + 551 \) is 76,176.
Solution
To find the sum \( 1 + 3 + 5 + \cdots + 551 \) using the formula \( S = n^2 \), follow these steps:
1. **Determine the number of terms (\( n \)):**
The last term of the sequence is 551. According to the hint:
\[
n = \frac{\text{last term} + 1}{2} = \frac{551 + 1}{2} = \frac{552}{2} = 276
\]
2. **Calculate the sum using the formula \( S = n^2 \):**
\[
S = 276^2 = 76176
\]
So, the sum of the series is:
\[
1 + 3 + 5 + \cdots + 551 = 76176
\]
**Answer:**
\( 1+3+5+\cdots+551=76176 \)
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Simplify this solution The Deep Dive
To find the sum of the series \( 1+3+5+\cdots+551 \), we first need to determine the value of \( n \). The last term in the series is 551. According to the hint, we add 1 to the last term and divide by 2: \[ n = \frac{551 + 1}{2} = \frac{552}{2} = 276 \] Now that we have \( n = 276 \), we can use the formula for the sum of the first \( n \) odd numbers, \( S = n^{2} \). So, \[ S = 276^{2} = 76176 \] Thus, the sum \( 1 + 3 + 5 + \cdots + 551 = 76176 \).
