91Ó°ÊÓ

The concept of the sum of a series involves adding the terms of a sequence to get a single value. Series can be finite or infinite. In our exercise, we have a finite series because it has a defined number of terms from 1 to 5. To find the sum, we add each term together.
Let's revisit the series given in the problem: \(\textstyle \sum_{i=1}^{5}(i+3)\).
This notation tells us to sum all the values of the expression \(i+3\) as \(i\) ranges from 1 to 5.
We substitute each value of \(i\) into the expression and then sum the results:

• For \(i = 1\): \(1+3 = 4\)
• For \(i = 2\): \(2+3 = 5\)
• For \(i = 3\): \(3+3 = 6\)
• For \(i = 4\): \(4+3 = 7\)
• For \(i = 5\): \(5+3 = 8\)

Thus, the series is written out as \(4 + 5 + 6 + 7 + 8\). By adding these numbers, we get \(30\). So, the sum of the series \(\textstyle \sum_{i=1}^{5}(i+3)\) is \(30\).

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In our series, the expression \(i+3\) is algebraic because \(i\) is the variable and \(+3\) is a constant added to \(i\).
The algebraic expression changes its value based on the value of the variable. By substituting different values for \(i\), we get different terms to sum:

• For \(i = 1\), the expression \(1+3\) becomes 4.
• For \(i = 2\), the expression \(2+3\) becomes 5.
• For \(i = 3\), the expression \(3+3\) becomes 6.
• For \(i = 4\), the expression \(4+3\) becomes 7.
• For \(i = 5\), the expression \(5+3\) becomes 8.

These values are then added together to form the sum of the series. This demonstrates how algebraic expressions come into play in series summation.

Summation notation is a concise way to write the sum of a sequence of terms. It is often represented with the Greek letter sigma \(\Sigma\). In the given exercise, \(\sum_{i=1}^{5}(i+3)\) uses summation notation to describe adding the terms of \(i+3\) as \(i\) ranges from 1 to 5.
Breaking down the notation:

• \(\Sigma\) signifies summation of the terms.
• \(i=1\) is the lower limit, indicating that \(i\) starts at 1.
• \(5\) is the upper limit, indicating that \(i\) ends at 5.
• The expression \(i+3\) is what we are summing for each value of \(i\).

Using summation notation makes it easier to represent and manipulate large sets of terms without writing them all out individually. For example, instead of writing 5 different terms and adding them individually, summation notation provides a streamlined way to understand and perform the summing operation.