Problem 61 - A Graphical Approach to College Algebra

Question

Evaluate each sum. $$\sum_{j=1}^{10}(2 j+3)$$

Step-by-Step Solution

Answer

The sum is 140.

Step 1: Understand the Problem

We are asked to evaluate the sum \( \sum_{j=1}^{10}(2j+3) \). This means we need to find the total by adding up the values of \(2j+3\) for each integer \(j\) from 1 to 10.

Step 2: Write the First Few Terms

Let's identify the first few terms of the sequence by substituting \(j = 1, 2, 3, \dots, 10\) into \(2j + 3\). For example, when \(j=1\), the term is \(2(1) + 3 = 5\). When \(j=2\), the term is \(2(2) + 3 = 7\). Continue this for \(j=3, 4, \ldots, 10\).

Step 3: Calculate Each Term

Evaluate each term in the sequence:- When \(j=1\), he term is \(5\).- When \(j=2\), the term is \(7\).- When \(j=3\), the term is \(9\).- When \(j=4\), the term is \(11\).- When \(j=5\), the term is \(13\).- When \(j=6\), the term is \(15\).- When \(j=7\), the term is \(17\).- When \(j=8\), the term is \(19\).- When \(j=9\), the term is \(21\).- When \(j=10\), the term is \(23\).

Step 4: Add the Terms

Sum all the terms together: \(5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23\).

Step 5: Simplify the Sum

Perform the addition step-by-step:- First pair and add: \(5+23=28\), \(7+21=28\), \(9+19=28\), \(11+17=28\), \(13+15=28\).- Sum these results: \(28\times5 = 140\).

Step 6: Write the Final Answer

The sum \(\sum_{j=1}^{10}(2j+3)\) is \(140\).

Key Concepts

Summation Notation

Summation notation is a concise way of expressing the sum of a sequence of numbers. In mathematics, it is represented by the Greek letter sigma (\(\Sigma\)). This symbol is followed by an expression of the terms you want to sum. For example, \(\sum_{j=1}^{10}(2j+3)\) is asking us to calculate the sum of the formula \(2j+3\) where \(j\) takes on all integer values from 1 to 10.

Summation notation has three parts:

• The lower limit of summation: This is the value where your sequence starts, such as \(j=1\).
• The upper limit of summation: This is where your sequence stops, like \(j=10\).
• The expression: What you are calculating for each successive value of \(j\), here it's \(2j+3\).

In our example, it states that for each value of \(j\) from 1 to 10, compute the expression \(2j+3\) and find the total of all these computed values. By using summation notation, complex descriptions and lengthy additions become more manageable and organized.

Sequence Calculation

The core task in evaluating a sum using summation notation involves calculating terms in a sequence. A sequence is simply a list of numbers in a particular order. In our example, each term in the sequence is given by the formula \(2j+3\), where \(j\) takes values from 1 to 10.

To calculate a sequence:

• Substitute each successive value of \(j\) into the expression \(2j+3\).
• For example, when \(j=1\), the term is \(2(1)+3=5\), and when \(j=2\), the term equals \(2(2)+3=7\).

The sequence produced from the expression would be: 5, 7, 9, 11, 13, 15, 17, 19, 21, 23.

Each term results from plugging the values between 1 and 10 into the expression. Recognizing these individual terms helps in understanding how the sequence progresses and prepares us for finding the sum of the sequence.

Series Evaluation

Evaluating a series means finding the sum of its sequence. A series is essentially an expression for the sum of a sequence of numbers. In our example, the series is written through summation notation: \(\sum_{j=1}^{10}(2j+3)\).

For series evaluation:

• Calculate each term in the sequence first, as we've done previously, noting them down.
• Example terms are: 5, 7, 9, ..., 23.
• Proceed to add these numbers together to find the total sum.

Here's a simple trick: if possible, pair terms from the start and end of the sequence since they often simplify faster.

For instance, pair 5 with 23, 7 with 21, and continue like that, each pair summing to 28 in this problem. Then multiplying the number of pairs (5) by the common pair sum (28) gives us the final answer, 140. This step-by-step formulation allows an organized and straightforward addition, leading us to the evaluated series sum.