Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Find the sum of each series. $$\sum_{j=1}^{15}(5 j-9)$$

Step-by-Step Solution

Answer

The sum of the series is 465.

Step 1: Understand the Series

The given series is \( \sum_{j=1}^{15}(5j-9) \). It means we have to calculate the sum of the expression \(5j - 9\) where \(j\) takes integer values from 1 to 15.

Step 2: Simplify the Expression

We need to find the sum of the series by calculating each term. The general term is \(5j - 9\). For each integer \(j\) from 1 to 15, this expression gives the term in the series.

Step 3: Calculate Each Term of the Series

Calculate each term:- For \(j = 1\), the term is \(5(1) - 9 = -4\).- For \(j = 2\), the term is \(5(2) - 9 = 1\).- Continue this process until \(j = 15\) to calculate all terms.

Step 4: Use the Formula for the Sum of an Arithmetic Series

This series is actually arithmetic with the first term \(a = 5 \times 1 - 9 = -4\) and the difference \(d = (5 \times 2 - 9) - (5 \times 1 - 9) = 5\). The formula for the sum of the first \(n\) terms of an arithmetic series is \(S_n = \frac{n}{2} (2a + (n-1)d)\).

Step 5: Substitute Values into the Formula

Substitute \(n = 15\), \(a = -4\), and \(d = 5\) into the sum formula:\[ S_{15} = \frac{15}{2} (2(-4) + (15-1) \times 5) \]Calculate each part.

Step 6: Simplify the Expression

Simplify within the formula:- Calculate \(2(-4) + 14 \times 5 = -8 + 70 = 62\).- Then, calculate \(\frac{15}{2} \times 62 = 465\).

Step 7: Conclude the Calculation

The sum of the arithmetic series \( \sum_{j=1}^{15}(5j-9) \) is 465. Each step in the calculation confirms this result.

Key Concepts

Series Sum

When dealing with series, particularly arithmetic series, it's essential to understand the concept of a sum. The **series sum** refers to the total value you get when adding up all the terms in the series. In mathematics, a series is the sum of the terms of a sequence. Think of a sequence as a list of numbers, whereas a series combines them through addition.
In our problem, the series is defined by the expression \(\sum_{j=1}^{15}(5j-9)\). Each term of the series is generated by plugging integers from 1 to 15 into the expression \(5j-9\). The aim is to sum these terms together.

• This sum tells us what the result is when each calculated term from the series is combined together.
• By systematically finding each term and adding them, we arrive at the series sum.

Calculating the series sum provides insight into how the sequence of numbers behaves and changes. This forms the backbone of solving many problems related to sequences in arithmetic.

Arithmetic Sequence

An **arithmetic sequence** is a sequence of numbers in which the difference of any two successive members is a constant. Each term in this sequence increases or decreases by the same amount, known as the common difference.
For the sequence \(5j-9\) as \(j\) varies from 1 to 15, we observe that the first term is \(-4\), obtained by substituting \(j=1\) into \(5j-9\). Every following term is achieved by increasing \(j\) by 1, thus forming an arithmetic progression.

• The common difference \(d\) in this particular sequence is 5, calculated from consecutive terms.
• This sequence characteristic allows us to use formulas to find the sum efficiently rather than adding each term individually.

Understanding this regular pattern is crucial because it simplifies computation and provides a pathway to the sum formula.

Sum Formula

The **sum formula** for an arithmetic series is an effective tool that allows for quick calculations without summing each term manually. The formula provides a direct method to find the total of terms in an arithmetic sequence.
The specific formula for the sum of the first \(n\) terms of an arithmetic sequence is:\[S_n = \frac{n}{2} (2a + (n-1)d)\]
Where:

• \(n\) is the total number of terms, here it is 15.
• \(a\) is the first term of the sequence, which we've found to be \(-4\).
• \(d\) is the common difference between the terms, which is 5 in this case.

Substituting the exercise's specific values into this formula, we see:\[S_{15} = \frac{15}{2} (2(-4) + (15-1) \times 5)\]Solving it step-by-step yields the sum 465. This formula reduces what could be an extensive process into something easy and manageable, emphasizing the elegance of patterns in arithmetic progressions.