Question
Find the sum. \( \sum_{k=1}^{4} 3 k^{2}=\square \)
Ask by Willis Chen. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Calculate the sum \( 3(1)^2 + 3(2)^2 + 3(3)^2 + 3(4)^2 = 3 + 12 + 27 + 48 = 90 \).
Solution
Calculate or simplify the expression \( \sum_{k=1}^{4} 3k^{2} \).
Find the sum by following steps:
- step0: Solution:
\(\sum _{k=1}^{4}3k^{2}\)
- step1: Calculate:
\(3\times \sum _{k=1}^{4}k^{2}\)
- step2: Use the formula \(\frac{n(n+1)(2n+1)}{6}:\)
\(3\times \frac{4\left(4+1\right)\left(4\times 2+1\right)}{6}\)
- step3: Multiply the numbers:
\(3\times \frac{4\left(4+1\right)\left(8+1\right)}{6}\)
- step4: Add the numbers:
\(3\times \frac{4\times 5\left(8+1\right)}{6}\)
- step5: Add the numbers:
\(3\times \frac{4\times 5\times 9}{6}\)
- step6: Multiply the terms:
\(3\times \frac{180}{6}\)
- step7: Divide the terms:
\(3\times 30\)
- step8: Multiply the numbers:
\(90\)
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Simplify this solution Extra Insights
To find the sum \( \sum_{k=1}^{4} 3 k^{2} \), we first calculate the individual terms by substituting the values of \( k \): - For \( k=1 \): \( 3(1^2) = 3 \times 1 = 3 \) - For \( k=2 \): \( 3(2^2) = 3 \times 4 = 12 \) - For \( k=3 \): \( 3(3^2) = 3 \times 9 = 27 \) - For \( k=4 \): \( 3(4^2) = 3 \times 16 = 48 \) Now, adding these terms together: \[ 3 + 12 + 27 + 48 \] Calculating step-by-step: - \( 3 + 12 = 15 \) - \( 15 + 27 = 42 \) - \( 42 + 48 = 90 \) Therefore, the sum is: \[ \sum_{k=1}^{4} 3 k^{2} = 90 \] Thus, \( \square = 90 \).
