91Ó°ÊÓ

In an arithmetic series, the common difference is the constant amount that each term in the series increases by when moving from one term to the next. To find it, simply subtract the first term from the second term. For example, in the arithmetic series given in the exercise, the difference between the first two terms is calculated as follows:

\[ d = a_2 - a_1 = 9 - 5 = 4 \]

Here, the common difference \( d \) is 4. This implies that each term is greater than the previous term by 4. Understanding the role of common difference is crucial when identifying patterns and calculating properties within arithmetic series, such as the total sum or the nth term.

The number of terms in an arithmetic series is not always explicitly given. However, it can be determined using the first term \( a_1 \), the last term \( a_n \), and the common difference \( d \). The formula is written as:

\[ n = \frac{a_n - a_1}{d} + 1 \]

By applying the formula to our problem, with the last term being 401 and the common difference being 4:

\[ n = \frac{401 - 5}{4} + 1 \]
\[ n = \frac{396}{4} + 1 \]
\[ n = 99 + 1 \]
\[ n = 100 \]

So, there are 100 terms in the series. Determining the number of terms is an essential step when attempting to find the sum of the series or when trying to identify specific terms within it.

Once the common difference and the number of terms are known, the sum of an arithmetic series can be calculated using the arithmetic series formula:

\[ \text{Sum} = \frac{n(a_1 + a_n)}{2} \]

where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term. For our exercise, plugging in the values we get:

\[ \text{Sum} = \frac{100(5 + 401)}{2} \]
\[ \text{Sum} = \frac{100(406)}{2} \]
\[ \text{Sum} = 50(406) \]
\[ \text{Sum} = 20300 \]

The sum of the series is 20300, which is reached by adding up all 100 terms starting from 5 and ending at 401, with each term increasing by a common difference of 4. This formula provides a swift alternative to adding each term individually, which is particularly useful for series with a large number of terms.