Problem 9 - Algebra for College Students

Question

Find the sum of each series. $$\sum_{j=0}^{5}(2 j-1)$$

Step-by-Step Solution

Answer

The sum of the series is 24.

Step 1: Identify the series elements

The given series is \(\text{sum}_{j=0}^{5}(2j-1)\). To find the sum, we first identify the elements of the series by substituting the values of \(j\) from 0 to 5.

Step 2: Substitute values into the series expression

Substitute \(j\) values into the expression \((2j-1)\).\When \(j = 0: 2(0) - 1 = -1\)\When \(j = 1: 2(1) - 1 = 1\)\When \(j = 2: 2(2) - 1 = 3\)\When \(j = 3: 2(3) - 1 = 5\)\When \(j = 4: 2(4) - 1 = 7\)\When \(j = 5: 2(5) - 1 = 9\)\Thus, the series elements are: \([-1, 1, 3, 5, 7, 9]\)

Step 3: Sum the series elements

To find the sum of the series elements, add them together: \((-1) + 1 + 3 + 5 + 7 + 9\).

Step 4: Calculate the total sum

Perform the addition from left to right:\((-1 + 1) = 0\)\((0 + 3) = 3\)\((3 + 5) = 8\)\((8 + 7) = 15\)\((15 + 9) = 24\)

Key Concepts

arithmetic series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. For example, in the series \([-1, 1, 3, 5, 7, 9]\), each number increases by 2 from the previous one, making it an arithmetic series.
Understanding this type of series is important because it allows you to find the sum quickly once you know a few components. Unlike random numbers, arithmetic series have a predictable pattern, which you can use to your advantage.

series sum calculation

Calculating the sum of an arithmetic series involves adding all the terms. Although this can be done directly, there are formulas to speed up the process. The formula for the sum of the first \(n\) terms of an arithmetic series is:

\[ S_n = \frac{n}{2} (a + l) \]
where:

• \( S_n \) is the sum of the first \( n \) terms.
• \( a \) is the first term.
• \( l \) is the last term.
• \( n \) is the number of terms.

This formula makes it easy to calculate the sum without adding each term individually.
In our earlier example, breaking down the series gives you a total sum of 24.

step-by-step solution

The step-by-step approach helps demystify complex problems by breaking them into smaller, manageable pieces. Let's revisit the steps from our example:

• Step 1: Identify the series elements
  The given series is \[ \text{sum}_{j=0}^{5}(2j-1) \]. We substitute each value of \( j \) from 0 to 5.


• Step 2: Substitute values into the series expression
  Substituting these values, we get the series elements \[ [-1, 1, 3, 5, 7, 9] \].


• Step 3: Sum the series elements
  Add each element: \( (-1) + 1 + 3 + 5 + 7 + 9 \)


• Step 4: Calculate the total sum
  Total = 24.

This clear and methodical approach makes solving these problems easier and helps in understanding the core concepts better.