91Ó°ÊÓ

An arithmetic sequence is a sequence of numbers in which the difference between any two successive terms is constant. This difference is known as the "common difference." In our exercise, the sequence is defined by the terms \(2z, 4z, 6z, \ldots, 20z\). Here, each term increases by \(2z\), making \(2z\) the common difference.
To identify an arithmetic sequence, look for this consistent pattern between terms. In our example, the initial term is \(2z\) and subsequent terms follow the pattern of adding \(2z\).
The general formula for any arithmetic sequence is given by the expression \(a_n = a_1 + (n-1)d\), where \(a_n\) represents the nth term in the sequence, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

• The first term \(a_1\) is \(2z\).
• The common difference \(d\) is \(2z\).

Understanding this helps simplify arithmetic sequences, enabling easier conversions to summation notation and subsequent calculations for sums.

To effectively find the sum of an arithmetic sequence, we need to understand how to sum a series of integers. In our specific exercise, the sum involves multiplying integers by a constant (\(2z\)).
When asked to find the sum of integers from 1 to a number \(n\), a well-known formula can be employed: the sum of the first \(n\) integers is \((n(n+1))/2\). For instance, in our exercise, we need the sum of the first 10 integers.
Here's how we arrive there:

• The formula is \(\sum_{n=1}^{n} n = \frac{n(n+1)}{2}\).
• Substitute \(n = 10\) into the formula to get \((10 \times 11)/2 = 55\).

This result is crucial because it forms part of the solution. We then multiply the sum by other constants involved in the sequence representation to find the final sum.

Summation notation offers a concise way to express the sum of a series or sequence. It uses the Greek letter sigma (\(\Sigma\)) to reduce lengthy addition into a manageable expression. In our exercise, the arithmetic sequence is compactly represented using summation notation.
The summation expression for the exercise is \(\sum_{n=1}^{10} 2nz\), which means you are summing up the terms \(2nz\) as \(n\) varies from 1 to 10. This notation helps succinctly handle sequences and series, making calculations easier by focusing on systematic patterns rather than individual terms.
The simplification relies on separating constants, in this case by factoring out \(2z\), to leverage the formula for the sum of integers. Hence, the summation formula not only simplifies expression but also calculation:

• The generic summation expression: \(\sum_{n=1}^{10} 2n = 2 \sum_{n=1}^{10} n = 2 \times 55 = 110\).
• Thus, multiply by \(z\) for the final sum: \(110z\).

By mastering both notation and strategy within summation calculations, you enhance your ability to effectively solve related problems using streamlined processes.