Problem 54 - Contemporary Precalculus

Question

$$\text { Find the sum.}$$ $$\sum_{n=1}^{30} \frac{4-6 n}{3}$$

Step-by-Step Solution

Answer

Answer: The sum of the first 30 terms of the given arithmetic series is -285.

Step 1: Identify the first term

To find the first term, substitute the first value of \(n\) (which is 1) into the given series: $$a_1 = \frac{4-6(1)}{3} = \frac{-2}{3}$$

Step 2: Identify the common difference

The common difference is the difference between two consecutive terms in an arithmetic series. Let's find the second term (by substituting \(n=2\)) and subtract the first term from it. $$a_2 = \frac{4-6(2)}{3} = \frac{-8}{3}$$ So the common difference is: $$d = \frac{-8}{3} - \frac{-2}{3} = -\frac{6}{3} = -2$$

Step 3: Use the arithmetic series summation formula

The arithmetic series summation formula is: $$S_n = \frac{n(2a_1+(n-1)d)}{6}$$ Plug in the values we found for n, \(a_1\), and d: $$S_{30} = \frac{30(2(\frac{-2}{3})+(30-1)(-2))}{6}$$

Step 4: Simplify the expression

Let's simplify the expression step by step: $$S_{30} = \frac{30(-\frac{4}{3}+(-2)(29))}{6}$$ $$S_{30} = \frac{30(-\frac{4}{3}-58)}{6}$$ $$S_{30} = \frac{30(-\frac{4+174}{3})}{6}$$ $$S_{30} = \frac{30(\frac{-170}{3})}{6}$$ $$S_{30} = \frac{-170\cdot 30}{3\cdot 6}$$ $$S_{30} = \frac{-5100}{18}$$ $$S_{30} = -285$$ The sum of the series is -285.

Key Concepts

Arithmetic Progression

An arithmetic progression (AP), also known as an arithmetic sequence, is a sequence of numbers in which the difference between any two consecutive terms is always the same. This difference is known as the "common difference." APs are quite common in various math problems because they simplify calculations involving series of numbers. These sequences can be increasing, if the common difference is positive, or decreasing, if the common difference is negative.

Consider a simple example: 2, 4, 6, 8, ... Here, the difference between each term is 2. This means it's an arithmetic progression with a common difference of 2.

• A general form for an AP is: \(a, a+d, a+2d, a+3d, \ldots\).
• Here, \(a\) is the first term, and \(d\) is the common difference.
• The n-th term of an arithmetic progression can be found using the formula: \( a_n = a + (n-1) \cdot d \).

By understanding these basic properties, tackling problems involving arithmetic progressions becomes much easier.

Summation Formula

An arithmetic series is the sum of the terms of an arithmetic progression. To find the sum quickly, we use a special formula. This is where the summation or series formula comes into play.

The formula to find the sum of the first \( n \) terms of an arithmetic series is:

• \[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \]
• Alternatively, if the last term isn't known, you can use: \[ S_n = \frac{n}{2} \cdot (2a_1 + (n-1) \cdot d) \]

In these formulas:

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

In our exercise, we calculated the sum \( S_{30} = -285 \) using these principles, highlighting the efficiency of this formula for large series.

Common Difference

The common difference in an arithmetic progression is a critical element that defines the spacing between consecutive terms. You always subtract the previous term from the next term to find it.

In our example from the exercise, we calculated the common difference \( d \) as follows:

• First Term (\( a_1 \)): \( \frac{-2}{3} \)
• Second Term (\( a_2 \)): \( \frac{-8}{3} \)
• Common Difference (\( d \)): \( d = a_2 - a_1 = \frac{-8}{3} - \frac{-2}{3} = -2 \)

This difference helps determine each subsequent term in the sequence and is central to calculating both individual terms and the overall sum of the series. Knowing the common difference can also help forecast the behavior of the sequence, whether it is increasing, decreasing, or constant.

First Term

The first term of an arithmetic progression is where the sequence starts. It sets the initial point for any arithmetic sequence and is denoted as \( a_1 \). Knowing the first term is essential because it, combined with the common difference, determines the entire sequence.

For instance, in our exercise, we determined the first term by substituting \( n=1 \) into the sequence formula:

• \[ a_1 = \frac{4-6(1)}{3} = \frac{-2}{3} \]

The first term often serves as a base for further calculations, such as predicting subsequent terms or calculating the overall sum using the summation formula. Understanding its role ensures a solid grasp of arithmetic progression fundamentals.