Problem 4 - College Algebra with Modeling and Visualization

Question

Write a series that represents the sum of the first seven positive odd integers, Find its sum.

Step-by-Step Solution

Answer

The sum is 49.

Step 1: Identify Odd Integers

The first seven positive odd integers are the numbers that fit the form \(2n - 1\) where \(n\) is a positive integer. Thus, the first seven positive odd integers are 1, 3, 5, 7, 9, 11, and 13.

Step 2: Write the Series

Express the sum of these numbers as a series: \[ S = 1 + 3 + 5 + 7 + 9 + 11 + 13 \]

Step 3: Use Summation Formula

Notice that the series is an arithmetic sequence where the first term \(a = 1\), the number of terms \(n = 7\), and the common difference \(d = 2\). The sum of an arithmetic series can be calculated using the formula: \[ S_n = \frac{n}{2} \times (a + l) \] where \(l\) is the last term of the series. In this case, \(l = 13\).

Step 4: Calculate the Sum

Substitute the values into the formula:\[ S_7 = \frac{7}{2} \times (1 + 13) = \frac{7}{2} \times 14 = 7 \times 7 = 49 \]

Step 5: Final Result

The sum of the first seven positive odd integers is 49.

Key Concepts

Positive Odd Integers

Positive odd integers are numbers that are not divisible by 2 and are greater than zero. They can be represented by a simple formula: \(2n - 1\), where \(n\) is a positive integer. This formula gives us a way to easily generate odd numbers by plugging in successive values of \(n\).

For example, when \(n = 1\), \(2(1) - 1 = 1\), which is the first positive odd integer. When \(n = 2\), \(2(2) - 1 = 3\), the second odd integer, and so on. Using this formula, you can list the first seven odd integers as found in the solution: 1, 3, 5, 7, 9, 11, and 13. Understanding this pattern is crucial when working with sequences, series, and more complex mathematical concepts.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms remains consistent. This difference is known as the 'common difference' and is denoted by \(d\). In the case of positive odd integers, the common difference is 2.

Consider the sequence 1, 3, 5, 7, 9, 11, and 13 from our exercise. Notice how each number is 2 more than the previous one. That's the common difference in action. Arithmetic sequences are easy to identify and analyze because of this regularity, which makes calculations like summations straightforward.

Summation Formula

The summation formula for arithmetic sequences helps find the sum of terms in the sequence quickly. The formula is given by

\[ S_n = \frac{n}{2} \times (a + l) \]

where \(S_n\) is the sum of the sequence, \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term.

Using our sequence of positive odd integers as an example, we apply:

• The first term \(a = 1\),
• The last term \(l = 13\), and
• The number of terms \(n = 7\).

Thus, substituting these values gives:
\[ S_7 = \frac{7}{2} \times (1 + 13) = \frac{7}{2} \times 14 = 7 \times 7 = 49 \]

This formula is powerful because it simplifies the process of adding a series of numbers, especially when applied to arithmetic sequences. By relying on this structured approach, you save time and reduce errors compared to adding each number individually.