Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Find the sum of each series. $$\sum_{i=1}^{3}(i+4)$$

Step-by-Step Solution

Answer

The sum is 18.

Step 1: Understand the Series

The given series is \( \sum_{i=1}^{3}(i+4) \). This represents the sum of a sequence where \( i \) takes values from 1 to 3, and for each \( i \), we calculate \( (i+4) \).

Step 2: Calculate Each Term

Substitute the values of \( i \) from 1 to 3 into the expression \( (i + 4) \).- For \( i = 1 \): \( 1 + 4 = 5 \)- For \( i = 2 \): \( 2 + 4 = 6 \)- For \( i = 3 \): \( 3 + 4 = 7 \)

Step 3: Sum the Terms

Add up the terms calculated in the previous step: \( 5 + 6 + 7 \).

Step 4: Final Calculation

Perform the addition: \( 5 + 6 = 11 \), then \( 11 + 7 = 18 \). The sum of the series is 18.

Key Concepts

Arithmetic Series

An arithmetic series is simply the sum of the terms in an arithmetic sequence. In an arithmetic sequence, each term after the first is found by adding a constant difference. This constant difference is known as the "common difference."
Understanding arithmetic series is critical because they frequently appear in mathematics, especially in algebra and calculus.

• Common difference: The difference between consecutive terms. For example, in our arithmetic sequence that generates the series, every term increases by 1 as we move from 5 to 6 and then from 6 to 7.
• First term: This is the starting point of the sequence. In our example, after substituting the initial value of the variable, the first term becomes 5.
• Number of terms: This defines how many terms are being added in the series. In this case, we have three terms: 5, 6, and 7.

To find the sum, you multiply the number of terms by the average of the first and last terms. In the context of our exercise, you see that the succession is simple enough to sum without a formula, but having this foundation helps unravel more complex series.

Sequence

A sequence is a collection of numbers arranged in a specific order. Each element in a sequence is called a term. In our exercise, the sequence consists of modifying numbers with the expression \( i+4 \) from \( i = 1 \) to \( i = 3 \).

• Defined by Rules: The rule of establishing our sequence is adding 4 to each counting number starting from 1. This gives us the sequence 5, 6, 7.
• Order Matters: Remember, order is crucial in sequences. Changing the order would inherently change the sequence itself.

A sequence like ours is called a finite sequence due to its limited set of terms. While tackling sequences, especially in series contexts, pay keen attention to the given rules or patterns used to generate them. This understanding greatly aids in procedural series exercises and future explorations into infinite sequences.

Sigma Notation

Sigma notation is a convenient method to express the sum of a sequence. It is represented using the Greek capital letter \( \Sigma \), symbolizing summation. This notation succinctly encapsulates potentially lengthy expressions by introducing limits and a general format.

• Limits of Summation: These are usually two numbers written above and below the \( \Sigma \). They indicate the starting and ending indices of summation. In our case, 1 is the lower bound and 3 is the upper bound.
• General Term: This is the expression following the \( \Sigma \) which defines how each term in the sequence is constructed. In our example, the general term is \( i+4 \).

Sigma notation is powerful because it provides a clear, formal structure for summing sequences efficiently. Learning to properly interpret and compute series using sigma notation is an essential skill, especially useful in higher mathematics like calculus.