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Summation notation is a concise way to represent the sum of a sequence of terms. It involves the use of the Greek letter sigma \(\sum\), which stands for summation, followed by an expression that describes the terms being added. For example, in the exercise provided, we have \(\sum_{k=1}^{20}(3k-5)\). This indicates that we need to sum the expression \(3k-5\) as \(k\) varies from 1 to 20.
The lower limit of summation is the number below the sigma, here it is \(k=1\), indicating where to start. The upper limit, \(k=20\), indicates the end of the summation. This notation is incredibly useful because it allows us to succinctly express a potentially large sum without having to write out all individual terms.

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. The general formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a + (n-1)d\), where:

• \(a\) is the first term,
• \(d\) is the common difference,
• \(n\) is the term number.

In our problem, the terms of the sequence are generated by the expression \(3k - 5\). Here, the first term (when \(k=1\)) is \(-2\), and the second term (when \(k=2\)) is 1, giving a common difference \(d = 3\). Notice how each term increases by 3, confirming this is an arithmetic sequence.

The sum of an arithmetic sequence, sometimes called an arithmetic series, can be calculated using a specific formula. When you want to find the sum \(S\) of the first \(n\) terms, use:
\[ S = \frac{n}{2} (a + l) \] where:

• \(n\) is the number of terms,
• \(a\) is the first term,
• \(l\) is the last term.

For the series \(3k-5\), with \(k\) ranging from 1 to 20, we have 20 terms. The first and last terms are \(-2\) and 55, respectively. Plugging these into the sum formula, we get:
\[ S = \frac{20}{2} (-2 + 55) = 10 \times 53 = 530\]
This formula provides a quick way to find the sum without adding each term individually.

An algebraic expression is a combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. The expressions can be quite simple or involve multiple terms and operations, much like the expression \(3k - 5\) in the exercise.

Each term of this sequence involves substituting a value for \(k\) and performing arithmetic operations to find a specific term's value. In algebra, understanding how to manipulate these expressions helps in simplifying problems and finding solutions, such as calculating sums of sequences efficiently.

• Here, for instance, you substitute each successive integer from 1 to 20 into \(3k - 5\) to generate the series terms.
• This is often the first step in identifying the type of sequence or pattern.

Learning to handle such expressions is foundational in algebra and critical in tackling more complex mathematical problems.