Problem 13 - Algebra 2 and Trigonometry

Question

In \(9-18,\) use the given information to a. write the series in sigma notation, and b. find the sum of the first \(n\) terms. $$ a_{1}=2, d=\frac{1}{2}, n=15 $$

Step-by-Step Solution

Answer

The series in sigma notation is \( \sum_{k=1}^{15} (2 + \frac{k-1}{2}) \) and the sum is 82.5.

Step 1: Understanding the Series

The series given is an arithmetic sequence, characterized by the first term \( a_1 = 2 \) and common difference \( d = \frac{1}{2} \). An arithmetic sequence can be expressed as \( a_n = a_1 + (n-1) \cdot d \), where \( a_n \) is the \( n \)-th term.

Step 2: Write the Series in Sigma Notation

The general term for the sequence is \( a_n = 2 + (n-1) \cdot \frac{1}{2} = 2 + \frac{n-1}{2} \). Thus, the series in sigma notation is \( \sum_{k=1}^{15} (2 + \frac{k-1}{2}) \).

Step 3: Compute the Sum of the Series

The sum \( S_n \) of the first \( n \) terms of an arithmetic series can be calculated by \( S_n = \frac{n}{2} \times (a_1 + a_n) \). Here, \( n = 15 \) and \( a_{15} = 2 + \frac{14}{2} = 9 \). Thus, \( S_{15} = \frac{15}{2} \times (2 + 9) = \frac{15}{2} \times 11 = 82.5 \).

Key Concepts

Sigma Notation

Sigma notation is a convenient way to express the sum of a sequence of numbers, using the Greek letter \( \Sigma \). This notation is particularly useful for arithmetic sequences, which are presented as a series of terms. The sigma notation for the given arithmetic sequence, where the first term is \( a_1 = 2 \) and the common difference \( d = \frac{1}{2} \), is written as:

• \( \sum_{k=1}^{15} \left(2 + \frac{k-1}{2}\right) \)

The expression inside the sigma notation, \(2 + \frac{k-1}{2}\), represents each term in the sequence, calculated by substituting values of \( k \) from 1 to 15. This is a shorthand way of writing the addition of multiple terms of the sequence.
Sigma notation not only simplifies the representation of an arithmetic sequence but also sets the stage for calculating the total sum of its terms.

Common Difference

The common difference \( d \) in an arithmetic sequence is crucial because it determines how each subsequent term progresses from the previous one.
In this exercise, the common difference is \( \frac{1}{2} \). This means each term increases by \( \frac{1}{2} \) from the term before it.

• To find the next term, simply add \( \frac{1}{2} \) to the current term.
• This ensures all terms in the sequence line up perfectly according to this uniform rate of change.

Let's take the first term \( a_1 = 2 \) as an example: the second term becomes \( 2 + \frac{1}{2} \), which equals \( 2.5 \), and the third term would be \( 2.5 + \frac{1}{2} = 3 \).
Understanding this pattern helps in formulating the general expression of the terms, which is \( a_n = 2 + \frac{n-1}{2} \). This formula is pivotal in using sigma notation and calculating the sum.

Sum of Series

The sum of an arithmetic series is calculated by adding all the terms in the sequence together. For a given number of terms, this can be efficiently computed using a handy formula:

• \( S_n = \frac{n}{2} \times (a_1 + a_n) \)

Here, \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the \( n \)-th term.
In our scenario, \( n = 15 \), \( a_1 = 2 \), and the \( 15^{th} \) term is \( a_{15} = 9 \). So the sum \( S_{15} \) is calculated as:

• \( S_{15} = \frac{15}{2} \times (2 + 9) = 82.5 \)

This formula utilizes the average of the first and last term, multiplied by the number of terms, making it a quick and accurate way of determining the total addition of the sequence's terms.

First n Terms

Understanding how to calculate the first \( n \) terms of an arithmetic sequence is pivotal for summing the sequence.
The sequence starts with the first term \( a_1 \) and follows a regular interval determined by the common difference \( d \). Each term is derived by adding \( d \) to the previous term, making the sequence predictable and uniform.In a formal mathematical expression, the \( n \)-th term \( a_n \) is calculated by:

• \( a_n = a_1 + (n-1) \cdot d \)

Let's put this into perspective using our example:

• The first term, \( a_1 = 2 \)
• For \( n = 15 \), the \( 15^{th} \) term \( a_{15} = 2 + \frac{14}{2} = 9 \)

Recognizing the sequence pattern helps when using sigma notation and simplifies summing the total of the sequence comfortably.