Problem 8 - Algebra for College Students

Question

Find the sum of each series. $$\sum_{j=0}^{3}(j+1)^{2}$$

Step-by-Step Solution

Answer

The sum is 30.

Step 1: Understand the Series Notation

Identify the given series notation \(\sum_{j=0}^{3}(j+1)^{2}\) and understand that it represents the sum of terms generated by the function \((j+1)^{2}\) as \(j\) ranges from \(0\) to \(3\).

Step 2: Generate the Terms of the Series

Substitute the values of \(j\) from \(0\) to \(3\) into the function \((j+1)^{2}\).For \(j=0\): \( (0+1)^2 = 1^2 = 1 \)For \(j=1\): \( (1+1)^2 = 2^2 = 4 \)For \(j=2\): \( (2+1)^2 = 3^2 = 9 \)For \(j=3\): \( (3+1)^2 = 4^2 = 16 \)

Step 3: Sum the Generated Terms

Add the calculated terms together:\( 1 + 4 + 9 + 16 \)

Step 4: Simplify the Sum

Simplify the expression to find the sum: \( 1 + 4 + 9 + 16 = 30 \)

Key Concepts

Series Notation

When dealing with series notation, it's important to first understand what the symbols mean. The notation \(\text{{\sum}}\) is the summation symbol, which tells us to add up a series of terms. The expression given in this exercise is \(\text{{\sum}}_{j=0}^{3}(j+1)^{2}\). Here, the variable \(j\) starts at 0 and increments by 1 until it reaches 3.
The term \( (j+1)^{2}\) is the function that generates the terms of the series as we substitute different values of \(j\). This means:

• When \(j=0\), we get \( (0+1)^2 = 1^2 = 1 \).
• When \(j=1\), we get \( (1+1)^2 = 2^2 = 4 \).
• When \(j=2\), we get \( (2+1)^2 = 3^2 = 9 \).
• When \(j=3\), we get \( (3+1)^2 = 4^2 = 16 \).

By understanding series notation, we know that we are summing up these particular terms.

Sum of Terms

Next, we need to focus on the sum of the terms we generated. We have already calculated individual terms for each value of \(j\). These terms are

• 1
• 4
• 9
• 16

To find the total sum of the series, we simply add these individual terms together. So, the calculation would be:
\( 1 + 4 + 9 + 16 \).
Summing these values, we get
\( 1 + 4 = 5 \),
\( 5 + 9 = 14 \),
\( 14 + 16 = 30 \).
Therefore, the sum of the series is \( 30 \).

Algebra Steps

Finally, it's essential to break down the algebra steps to ensure everything adds up correctly. Here are the detailed steps we took:

• Generated terms using the function \( (j+1)^{2} \) for each value of \( j \) from 0 to 3.
• Computed the individual values: 1, 4, 9, and 16.
• Summed up these values step-by-step:
  \( 1 + 4 = 5 \)
  \(5 + 9 = 14 \)
  \(14 + 16 = 30\).

By following these clean algebra steps, we clearly see how the final sum of the series \( \text{{\sum}}_{j=0}^{3}(j+1)^{2} \) is calculated to be 30. This systematic approach ensures that we understand not just the final answer, but also how we arrived there.