Problem 1 - Calculus Single Variable

Question

P In Exercises \(1-10\), write out the sum, and perform the addition. $$ \sum_{j=1}^{4} 3 j $$

Step-by-Step Solution

Answer

The sum is 30.

Step 1: Understanding the Sum Notation

The given exercise involves a summation notation, \( \sum_{j=1}^{4} 3j \), which means that we need to evaluate the sum of the expression \( 3j \) as \( j \) varies from 1 to 4. This indicates there's a series of terms generated by substituting the integer values of \( j \) from 1 through 4 into the expression \( 3j \).

Step 2: Substituting Values Into the Expression

Substitute the values of \( j \) from 1 to 4 into the expression \( 3j \):- For \( j = 1 \), the term is \( 3 \times 1 = 3 \).- For \( j = 2 \), the term is \( 3 \times 2 = 6 \).- For \( j = 3 \), the term is \( 3 \times 3 = 9 \).- For \( j = 4 \), the term is \( 3 \times 4 = 12 \).

Step 3: Writing Out the Sum

Write out the terms as a sum: \( 3 + 6 + 9 + 12 \).

Step 4: Performing the Addition

Add up the terms obtained: - First, add \( 3 + 6 = 9 \).- Then add \( 9 + 9 = 18 \) (adding 9 to the previous result).- Finally, add \( 18 + 12 = 30 \) to get the final result.

Key Concepts

Understanding Series

A series is essentially the sum of the terms of a sequence. In mathematics, it is crucial for identifying the total when a certain pattern or rule generates the numbers, which are then added together. For example, take a simple series created by adding numbers like 1, 2, 3, and so forth. When these numbers are added, it becomes a series like 1 + 2 + 3 = 6.

When working with series, remember that the addition follows a specific order from the first term to the last. It reflects a continuous cumulative process.

• The series can be finite or infinite based on the amount of terms included.
• Finite series have a definite number of terms, whereas infinite series continue indefinitely.
• The series in our step-by-step solution is finite and stops after four terms: 3, 6, 9, and 12.

This way of looking at series helps in simplifying problems by breaking down numbers step-by-step rather than handling complex arithmetic in one leap.

Exploring Arithmetic Sequences

An arithmetic sequence is a sequence of numbers where each term is generated by adding a constant to the previous term. This constant is called the "common difference." It is the same value added (or subtracted) throughout the sequence.

Consider, for example: 2, 5, 8, 11, ... In this sequence, each term is formed by adding a constant difference of 3 to the previous term. Arithmetic sequences are handy because they have a regular pattern which makes predicting terms straightforward.

• To find the general term (or any specific term) in an arithmetic sequence, we use the formula: \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.
• In our original exercise, the sequence generated by \( 3j \) follows an arithmetic sequence as well: 3, 6, 9, 12.
• Here, the common difference is 3, which is the multiplier of \( j \) in \( 3j \).

Understanding these sequences is crucial for mastering summation problems, as they often form the basis of the series being examined.

Demystifying Notation

Mathematics often uses a special language of symbols to simplify and communicate complex ideas efficiently. One such concept is summation notation, which is used to denote the addition of a series of numbers. It is typically represented by the sigma symbol \( \sum \).

In our exercise, the summation notation \( \sum_{j=1}^{4} 3j \) tells us to sum each instance of \( 3j \) from \( j = 1 \) to \( j = 4 \). This form of notation serves several purposes:

• It simplifies the expression of adding many terms by compactly representing a repetitive addition process.
• Using indices, it clearly defines the starting and ending points of the series or sequence being summed.
• This notation helps in visualizing the process of expanding the terms (here: from 3 to 12 by steps of 3).

Learning to read and write such notations is a foundational skill in mathematics, as it streamlines what could otherwise be cumbersome calculations.