Question

Evaluate the sums in Problems 5-12 using the formula for the sum of an arithmetic series. $$ -4.01-4.02-4.03-\cdots-4.35 $$

Short answer

Answer: The sum of the given arithmetic series is -146.3.

Step-by-step solution

Calculate the number of terms

First, we'll calculate the number of terms using the first term (\(a_1\)), last term (\(a_n\)), and common difference (\(d\)): $$ n = \frac{a_n - a_1}{d} + 1 $$ Substituting the given values: $$ n = \frac{-4.35 - (-4.01)}{-0.01} + 1 $$ $$ n = 35 $$ So there are 35 terms in the given arithmetic series.

Apply the formula for the sum of an arithmetic series

Now we can find the sum by applying the formula for the sum of an arithmetic series: $$ S_n = \frac{n}{2}(a_1 + a_n) $$ Substitute the values of \(n\), \(a_1\), and \(a_n\): $$ S_{35} = \frac{35}{2}(-4.01 - 4.35) $$ $$ S_{35} = \frac{35}{2}(-8.36) $$ $$ S_{35} = -146.3 $$ Therefore, the sum of the given arithmetic series is -146.3.

Key concepts

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. This constant difference between the numbers is known as the "common difference." The general form of an arithmetic sequence can be expressed as: - First Term (\(a_1\))- Second Term (\(a_2 = a_1 + d\)- Third Term (\(a_3 = a_2 + d\))
As you can see, each term after the first is formed by adding the common difference \(d\) to the previous term.
For the sequence in this exercise, the sequence starts from -4.01 and ends at -4.35, using a common difference of -0.01. This means every term in this sequence is 0.01 less than the previous term.

Sum Formula

The sum of an arithmetic series can be found with a specific formula. This formula is helpful when you need the sum of all the terms in the sequence. 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 first \(n\) terms,
• \(n\) is the number of terms,
• \(a_1\) is the first term of the series,
• \(a_n\) is the last term of the series.

By plugging these values into the formula, you can easily find the sum of the entire sequence. In the provided example, we used this formula to calculate that the sum of the series from -4.01 to -4.35 is -146.3.

Common Difference

The common difference is a crucial element of an arithmetic sequence. It is the constant amount added (or subtracted) to each term to reach the next term. The common difference \(d\) can be found by subtracting the first term from the second term:\[d = a_2 - a_1\]
In the exercise, the common difference is -0.01, meaning each term in the sequence is 0.01 less than its predecessor. Understanding the common difference helps in identifying how the sequence progresses and is essential for finding other parameters like the number of terms or the sum.

Number of Terms

Determining the number of terms in an arithmetic sequence is vital for correctly using the sum formula. You can find the number of terms using the following formula:\[n = \frac{a_n - a_1}{d} + 1\]
Where:

• \(n\) is the number of terms,
• \(a_1\) is the first term,
• \(a_n\) is the last term,
• \(d\) is the common difference.

In this example, by using -0.01 as the common difference, the first term -4.01, and the last term -4.35, we calculated there are 35 terms in the series. Knowing the number of terms is crucial to finding the series' sum and understanding its extent.