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An arithmetic sequence is a list of numbers in which each term after the first is obtained by adding a constant, called the 'common difference', to the previous term. This sequence appears in a linear pattern. The first term in our exercise is 4, and the second term is derived by adding the common difference to the first.

Let's consider the series given: 4, 9, 14, and so on, expressed as

• The first term, 4.
• Adding a common difference of 5 gives the next term, 9.
• Following the same principle, adding 5 again results in 14.

Thus, we can represent the general term in this arithmetic sequence as \[a_n = a + (n-1)d \] where \( a \) is the first term and \( d \) is the common difference. Understanding this helps in solving for terms and finding missing values in sequences.

When dealing with arithmetic sequences, a common task is to find the sum of a given number of terms. This is known as the sum of an arithmetic series. The sum can be calculated using a specific formula that simplifies the process greatly.

Given an arithmetic series, the sum of the first \( n \) terms, denoted \( S_n \), is calculated with the formula:\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]Here, \( a \) represents the first term, \( n \) is the number of terms, and \( d \) is the common difference between the terms. Using this formula makes it easy to find the sum without having to manually add each individual term. For instance, in our problem, substituting the known values ready to solve the expression can swiftly yield the desired total.

• First term (\( a \): 4)
• Common difference (\( d \): 5)
• Number of terms (\( n \): evaluated from the sequence equation)

The term 'common difference' is vital in any arithmetic sequence. It refers to the constant amount that we add (or subtract, if negative) to obtain successive terms in the sequence. In our given exercise, the common difference \( d \) is 5.

Here is how it works in our sequence:

• From the first term 4 to the second term 9, we add 5 (9 - 4 = 5).
• From the second term 9 to the third term 14, adding 5 again results in 14.

Thus, the common difference plays a significant role in determining the structure of the sequence. It helps in generating terms and finding the sequence's general term or even verifying the sequence's correctness. Without a consistent difference, a series cannot be classified as arithmetic.

Finding the number of terms in an arithmetic series is an essential step, especially when calculating the sum of the series. From the exercise, we learn how to determine \( n \), the number of terms, when given a specific condition for the last term.

Assume you have an arithmetic series with the last term expressed as \( 5n + 4 \). To find \( n \), you should equate the expression for the last term to the formula for any term in the sequence:\[ a + (n - 1)d = 5n + 4 \]Solve this equation for \( n \) to obtain the number of terms. This step is crucial for evaluating the entire series, as knowing how many terms exist ensures your sum calculation is accurate and complete. It is also used to derive further insights and understand the behavior of the series, as managed within the exercise review.