91Ó°ÊÓ

The heart of understanding an arithmetic series lies in the concept of the common difference. An arithmetic series is simply a sequence of numbers in which each term after the first is found by adding a constant value, known as the common difference, to the preceding term.

In the given exercise, the common difference is found by subtracting any term from the term that follows it. For example:
\[31 - 25 = 6\] and \[613 - 607 = 6\]
These calculations confirm that the series has a common difference (d) of 6. Identifying the common difference is crucial, as it is a vital component used to find both the sum of the series and the number of terms within it. In general, if the common difference is positive, the series will be increasing, and if it is negative, the series will be decreasing. Knowing this, you can better understand the nature and behavior of the series.

Once you have the common difference and the number of terms, you can find the sum of an arithmetic series using the summation formula. This invaluable tool is expressed as:
\[S_n = \frac{n}{2}(F + L)\]
Where \(S_n\) is the sum of the first n terms, \(F\) is the first term, \(L\) is the last term, and \(n\) is the total number of terms in the series.

In our exercise, with 99 terms, the first term being 25, and the last term being 613, we plug these values into the formula to calculate the sum:
\[S_{99} = \frac{99}{2}(25 + 613)\]
\[S_{99} = \frac{99}{2}(638)\]
\[S_{99} = 49.5 \cdot 638\]
\[S_{99} = 31599\]

Therefore, the sum of our arithmetic series is 31,599. Understanding and using this formula will allow you to quickly sum up a large number of terms without needing to add each term individually.

Determining the number of terms in an arithmetic series is a critical step towards finding its sum. You can calculate the number of terms using the formula:
\[n = \frac{(L - F)}{d} + 1\]
Where \(n\) is the number of terms, \(L\) is the last term, \(F\) is the first term, and \(d\) is the common difference. Intuitively, this formula tells us how many increments of the common difference it takes to get from the first term to the last term in the series.

In our example, with a last term of 613, a first term of 25, and a common difference of 6, the number of terms in the series is calculated as:
\[n = \frac{(613 - 25)}{6} + 1\]
\[n = \frac{588}{6} + 1\]
\[n = 98 + 1\]
\[n = 99\]

This tells us there are 99 terms in the series. Knowing the number of terms is essential not only for summing the series but also for understanding the structure and length of an arithmetic sequence. This concept is particularly helpful when dealing with large series, where manually counting each term is impractical.