Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Find the sum for each series. $$\sum_{i=1}^{100} 6$$

Step-by-Step Solution

Answer

The sum is 600.

Step 1: Understand the Problem

We are tasked with finding the sum of a repeating constant over a sequence. This sequence uses the index \(i\) that goes from 1 to 100, and at every step, the value of the sequence is 6.

Step 2: Identify the Formula

The formula for the sum of a constant \(c\) added \(n\) times is given by \( \, c \times n \, \). Here, the constant is 6, and the number of terms \(n\) is 100.

Step 3: Calculate the Sum

Substitute \(c = 6\) and \(n = 100\) into the formula \(c \times n\). This gives us \(6 \times 100\).

Step 4: Compute the Result

Perform the multiplication \(6 \times 100\) to get the total sum. The result is 600.

Key Concepts

Arithmetic Series

An arithmetic series is a sum of terms in an arithmetic sequence. It is a sequence where each term is derived by adding a fixed number, known as the common difference, to the previous term. For example, in the sequence 2, 4, 6, 8, the common difference is 2. In arithmetic series, we focus on the summation of terms:

• If the first term of the sequence is \(a\) and the common difference is \(d\), the \(n\)-th term can be expressed as \(a_n = a + (n-1) \cdot d\).
• The sum of the first \(n\) terms \(S_n\) of an arithmetic sequence is calculated using the formula: \(S_n = \frac{n}{2} (2a + (n-1) \cdot d)\).

Note that this exercise involves a constant term sequence, where the arithmetic series concept slightly differs as the common difference is zero.

Sequence

A sequence is an ordered list of numbers. Each number in the list is called a term. Sequences can be finite or infinite, depending on whether they have a last term. For example:

• An arithmetic sequence has a constant difference between consecutive terms.
• A geometric sequence, on the other hand, has a constant ratio between consecutive terms.

In this exercise, we deal with a simple form of sequence where each term is the same, a constant value of 6 repeated for 100 terms.

Summation Formula

The summation formula is a powerful tool in mathematics that allows for the concise calculation of the sum of a sequence. In contexts like this exercise, where the sequence is composed of a constant term, the summation becomes straightforward. To find the sum:

• Identify the constant term \(c\), which is 6 in this case.
• Count the number of terms \(n\), which here is 100.

The summation formula for a constant series is \(c \times n\). This formula simplifies the process by avoiding the need to manually add the constant value 100 times.

Constant Series

A constant series is perhaps the simplest type of series where all terms are the same. When calculating the sum of a constant series:

• The constant term is repeated across the range of the series.
• For example, in our problem, the constant term is 6.

To find the sum, we simply multiply this constant by the number of terms. Because all values are the same, the series is linear, making it very easy to compute the sum. In this case, multiplying 6 by the 100 terms gives us a total sum of 600.