Problem 14

Question

Express the sums in Exercises $11-16$ in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$ 2+4+6+8+10 $$

Step-by-Step Solution

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Answer

$\sum_{n=1}^{5} 2n$ represents the sum from 2 to 10 in sigma notation.

1Step 1: Identify the sequence pattern

First, observe the sequence given: $2, 4, 6, 8, 10$. Notice that this is an arithmetic sequence where each term increases by 2.

2Step 2: Determine the general term of the sequence

The sequence starts at 2 and follows the pattern where each term can be mathematically represented. The general term of an arithmetic sequence can be written as $a_n = a_1 + (n-1)d$, where $a_1 = 2$ and $d = 2$. Therefore, the term can be expressed as $a_n = 2n$.

3Step 3: Express the sequence in sigma notation

Using the general term $a_n = 2n$, the sequence can be expressed in summation (sigma) notation. The sequence has 5 terms. So, the sigma notation with summation limits is: $\sum_{n=1}^{5} 2n$.

Key Concepts

Understanding Arithmetic SequencesThe Power of Summation NotationConstructing the General Term

Understanding Arithmetic Sequences

An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a constant difference to the previous term. This constant difference is known as the "common difference." In the example given, the sequence is 2, 4, 6, 8, 10. Here, the common difference is 2 because each term increases by 2 from the one before it. Identifying the first term and the common difference is crucial in solving problems involving arithmetic sequences. In our example:

First Term ($a_1$): 2
Common Difference ($d$): 2

The simplicity of arithmetic sequences comes from their predictable nature. Once the first term and common difference are known, any term in the sequence can be calculated. This predictability makes it easy to express the series in a formulaic way, aiding in further mathematical expressions such as summation notation.

The Power of Summation Notation

Summation notation, often represented by the Greek letter sigma ($ \Sigma $), is a powerful mathematical tool that allows us to express the sum of a sequence of terms in a concise form. It specifies three main components: the general term, the variable of summation (typically $n$), and the limits of summation.For the sequence 2, 4, 6, 8, 10, we can express it in summation notation as $ \sum_{n=1}^{5} 2n $. Breaking it down:

Base of Sigma ($2n$): Represents the general term of the sequence.
Variable ($n$): Indicates the term number in the sequence.
Limits of Summation: From 1 to 5, because there are 5 terms in the sequence.

This notation is particularly useful in mathematics as it simplifies the expression of lengthy sequences and provides a unified way to deal with sums in advanced topics.

Constructing the General Term

The general term of an arithmetic sequence is a formula that allows us to find any term in the sequence without listing all of its preceding terms. For any arithmetic sequence, it's given by:\[a_n = a_1 + (n-1) \cdot d\]where:

$a_n$: The nth term of the sequence.
$a_1$: The first term of the sequence.
$d$: The common difference between consecutive terms.
$n$: The term number.

In the provided sequence, the first term $a_1$ is 2, and the common difference $d$ is also 2. Therefore, our general term becomes $a_n = 2 + (n-1) \times 2 = 2n$. Simply put, the formula says that any term in this sequence can be calculated by multiplying its position number $n$ by 2. Understanding how to derive and use the general term is essential when dealing with arithmetic sequences, as it forms the backbone for reasoning about their properties and for converting them into summation notation.

Problem 14

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