Question

Evaluate the sums in Problems 5-12 using the formula for the sum of an arithmetic series. $$ 22.5+20.5+18.5+\cdots+2.5+0.5 $$

Short answer

Answer: The sum of the arithmetic series is 138.

Step-by-step solution

Identify the first term (a) and the common difference (d)

The first term of the arithmetic series is \(a = 22.5\). The common difference, d, can be found by subtracting two consecutive terms: \( d = 20.5 - 22.5 = -2\).

Find the number of terms (n) in the series

To find the number of terms, n, we can use the formula: $$n = \frac{last\_term -first\_term}{d} +1$$, where last_term is the final term in the sequence, which in this case is 0.5. Thus: $$n = \frac{0.5 - 22.5}{-2} + 1 = \frac{-22}{-2} + 1 = 11 + 1 = 12$$ There are 12 terms in the series.

Use the sum formula for arithmetic series to calculate the sum

The sum formula for an arithmetic series is: $$S_n = \frac{n}{2}(a_1 + a_n)$$, where \(S_n\) is the sum of the series, n is the number of terms, and \(a_1\) and \(a_n\) are the first and last terms of the series, respectively. In our case, we have \(n=12\), \(a_1=22.5\), and \(a_n=0.5\). Plugging these values into the formula, we get: $$ S_n = \frac{12}{2}(22.5 + 0.5) = 6(23) = 6\cdot 23 = 138 $$ So, the sum of the given arithmetic series is 138.

Key concepts

Sum of a Series

The sum of a series, especially an arithmetic series, is a fundamental concept in mathematics. When we talk about the sum of an arithmetic series, we're referring to the total of all terms from the beginning to the end of that series. Arithmetic series have a special formula that makes this calculation straightforward:

• Formula: \[ S_n = \frac{n}{2}(a_1 + a_n) \]

This formula is remarkably efficient because, instead of adding each term step by step, it uses the shortcut of taking the average of the first and last terms and multiplying it by the number of terms.
For example, in the problem provided, we determined that there are 12 terms with a first term of 22.5 and a last term of 0.5. Plugging these into the formula gives us:

• \[ S_{12} = \frac{12}{2}(22.5 + 0.5) = 6 \times 23 = 138 \]

This efficient process allows us to quickly find the sum of even a lengthy arithmetic series.

Common Difference

The common difference in an arithmetic series is a crucial element that defines the pattern and progression of the series. This is the constant value that is added (or subtracted if negative) to each term to generate the next term in the sequence.
To find the common difference, denoted by \(d\), we simply subtract any term from the next one:

• Formula:\[ d = a_{n+1} - a_n \]For our series, the first few terms are 22.5, 20.5, 18.5, etc. Subtracting the first term from the second, we find:\[ d = 20.5 - 22.5 = -2 \]

This means each term in the series decreases by 2. A negative common difference indicates a decreasing series, which is apparent here as each succeeding number is less than the one before.
Understanding the common difference is key to determining both the behavior of the series and to finding other series attributes, such as the number of terms.

Number of Terms

Calculating the number of terms in an arithmetic series is essential for applying the sum formula correctly. To find this, we use a formula that involves the first term, the last term, and the common difference.
For any arithmetic series, the number of terms \(n\) can be found by:

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

In our example, the initial term \(a_1\) is 22.5, the last term \(a_n\) is 0.5, and the common difference \(d\) is -2. Plugging these values in, we get:

• \[ n = \frac{0.5 - 22.5}{-2} + 1 = \frac{-22}{-2} + 1 = 11 + 1 = 12 \]

This tells us there are 12 terms in the series. Knowing the number of terms is crucial, as it directly impacts the calculation of the series sum using the formula.