91Ó°ÊÓ

An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant number, called the common difference. Calculating the sum of an arithmetic series can be simple once you understand the formula. For an arithmetic series, the sum of the first \(n\) terms, represented as \(S_n\), is given by the formula:

• \(S_n = \frac{n}{2} (2a + (n-1)d)\)

Here, \(a\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms.
This formula essentially takes the average of the first and last term, multiplied by the number of terms, to find the total sum.
Understanding and applying this formula is key to solving not only this problem but others involving arithmetic series as well.

The common difference, represented by \(d\), is a pivotal concept in understanding arithmetic sequences. It is the number that is consistently added to each term to get the next term in the sequence.
In the exercise we are addressing, the common difference is found by subtracting the first term from the second term.

• In this case: \(19 - 7 = 12\)

This means that each term in the sequence is 12 units greater than the previous term.
Recognizing the common difference is essential, as it helps in finding any term in the sequence, as well as calculating the sum of the sequence.

The first term of an arithmetic sequence, denoted by \(a\), is the starting point of the sequence. It is crucial in determining the rest of the series.
In the given exercise, the first term \(a\) is 7.

• It acts as the initial baseline from which the entire sequence progresses by adding the common difference.

Once the first term and the common difference are known, any term in the sequence can be easily calculated using the formula for the \(n\)th term, \(a_n = a + (n-1)d\).

The number of terms, represented by \(n\), specifies how many terms of the sequence you're dealing with. It plays a key role in determining the sum of the arithmetic series.
Knowing \(n\), you can determine not only the full extent of the sequence you are considering but also utilize it directly in the sum formula.

• For the exercise, we are concerned with the first 25 terms, so \(n = 25\).

The greater the \(n\), the more terms there are to add up, which impacts the sum greatly, as each new term introduces more value to the total of the sequence.
Thus, always ensure to correctly identify \(n\) right from the start in such problems.