Question

Evaluate the sums. \(\sum_{i=1}^{4}(2+i)\)

Short answer

Answer: The value of the summation \(\sum_{i=1}^{4}(2+i)\) is 18.

Step-by-step solution

Understand the summation notation

The given summation notation is \(\sum_{i=1}^{4}(2+i)\), which means that we need to calculate each term in the sequence using the term formula, \((2+i)\), for each value of i between 1 and 4, inclusive, and then sum them all up.

Calculate the terms for i = 1, 2, 3, and 4

We will now calculate the terms for i = 1, 2, 3, and 4. For \(i = 1\), term = \((2 + 1) = 3\) For \(i = 2\), term = \((2 + 2) = 4\) For \(i = 3\), term = \((2 + 3) = 5\) For \(i = 4\), term = \((2 + 4) = 6\)

Sum the calculated terms

Now that we have the terms for i = 1, 2, 3, and 4, we can sum them up: Sum = \(3 + 4 + 5 + 6 = 18\) So, \(\sum_{i=1}^{4}(2+i) = 18\).

Key concepts

Summation Notation

Summation notation is a concise way of expressing the addition of a sequence of numbers. It is represented by the Greek letter sigma (\(\Sigma\)). This method is helpful to condense long sums into a more manageable form. In this system, we have an index of summation, here it is indicated by the variable \(i\), which represents each term's position in the sequence.

For example, the expression \(\sum_{i=1}^{4}(2+i)\) defines:

• The index \(i\) starts at 1 and ends at 4.
• The term formula, which is \((2 + i)\), determines the actual numbers we add together.
• The process involves substituting \(i\) in the term formula sequentially from 1 to 4.

Using this notation makes it simpler to represent and compute the total of multiple similar terms. Instead of writing each number you want to sum, you define the rule once and apply it across a range.

Sequence

A sequence is a list of numbers ordered logically or according to a specific rule. Each item in this list is an element of the sequence. The order can be finite or infinite, determined by certain conditions or boundaries.

In the context of the summation example \(\sum_{i=1}^{4}(2+i)\), the sequence is constructed as follows:

• The first term when \(i=1\) gives \(2+1 = 3\).
• The second term when \(i=2\) gives \(2+2 = 4\).
• The third term when \(i=3\) gives \(2+3 = 5\).
• The fourth term when \(i=4\) gives \(2+4 = 6\).

Sequences are fundamental in mathematics as they form the basis for more complex functions and series. This example results in the sequence \(3, 4, 5, 6\). Sequences can represent patterns, help solve problems, and are essential in various fields of study.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. An arithmetic sequence is a list of numbers with a constant difference between consecutive terms.

The arithmetic sequence from the summation \(\sum_{i=1}^{4}(2+i)\) is \(3, 4, 5,\) and \(6\). The difference between each term in this sequence is \(1\), making it arithmetic.

In an arithmetic series, we add up all these terms. One way to efficiently calculate the sum of an arithmetic series is by using the formula:

• Sum = \( \frac{n}{2} \times (\text{First Term} + \text{Last Term}) \)

Here, \(n\) is the number of terms. In our example, with terms \(3, 4, 5,\) and \(6\), the sum calculated is:

• Number of terms, \(n = 4\)
• First term is \(3\), and last term is \(6\)
• Sum = \( \frac{4}{2} \times (3 + 6) = 2 \times 9 = 18\)

This matches the calculated summation result, showing the connection between summation, sequences, and arithmetic series.