Problem 19 - Contemporary Precalculus

Question

Let \(c\) and \(d\) be fixed real numbers. Prove that $$\begin{aligned} c+(c+d)+(c+2 d) &+(c+3 d)+\cdots \\ &+[c+(n-1) d]=\frac{n[2 c+(n-1) d]}{2} \end{aligned}$$

Step-by-Step Solution

Answer

Question: Prove that the given equation holds true for fixed real numbers, \(c\) and \(d\): \(c+(c+d)+(c+2 d)+(c+3 d)+\cdots + [c+(n-1)d] = \frac{n[2 c +(n-1) d]}{2} \)

Step 1: Identify the common difference and the first term

The first step in this proof is to recognize that the given equation represents the sum of an arithmetic series. The common difference between the terms of the series is \(d\), and the first term is \(c\). We will be working with the equation: \(c+(c+d)+(c+2d)+(c+3d)+ \cdots + [c+(n-1)d]\)

Step 2: Identify the last term of the series

The last term of the series, often referred to as \(l\), is given by \(c+(n-1)d\). This expression represents the nth term of the arithmetic series.

Step 3: Apply the formula for the sum of an arithmetic series

The formula for the sum of an arithmetic series is: \(S_n = \frac{n(a+l)}{2}\) where \(S_n\) is the sum of the first n terms, \(a\) is the first term, \(l\) is the last term, and \(n\) is the number of terms.

Step 4: Substitute the values of the first term, last term, and the number of terms

Given the arithmetic series: \(c+(c+d)+(c+2d)+(c+3d)+ \cdots + [c+(n-1)d]\) we know that the first term, \(a\), is \(c\), and the last term, \(l\) is \(c+(n-1)d\). We also know that there are \(n\) terms in this series. We can substitute these values into the formula for the sum of an arithmetic series: $$S_n = \frac{n(c + [c + (n-1)d])}{2}$$ or, $$S_n = \frac{n(2c + (n-1)d)}{2}$$

Step 5: Conclude

With the substitution in Step 4, we arrive back at the equation which we are proving, namely: \(c+(c+d)+(c+2 d)+(c+3 d)+\cdots + [c+(n-1)d] = \frac{n[2 c +(n-1) d]}{2} \) Thus, we have proved that the given equation holds true for fixed real numbers, \(c\) and \(d\).

Key Concepts

Arithmetic Progression

An arithmetic progression (AP), also known as an arithmetic sequence, is one of the most commonly encountered sequences in mathematics. It consists of a series of numbers in which each term after the first is formed by adding a constant, called the common difference, to the previous term. For instance, if we start with a term, say 3, and continually add a common difference of 2, we get the sequence 3, 5, 7, 9, and so forth.

In our exercise, the sequence starts with the term 'c' and has a common difference of 'd'. The sequence can be written as \( c, c+d, c+2d, c+3d, ... \) where the nth term \( t_n \) is given by the formula \( c + (n - 1) * d \).

The concept of arithmetic progression is vital in understanding the behavior of evenly spaced elements in various mathematical contexts and real-world situations. This principle applies to time intervals, distances, and any setting where regular increments are present.

Sequence and Series

A sequence is a set of numbers listed in a specific order, following a certain rule that defines the pattern. Moreover, a series is the sum of the terms of a sequence. While the terms of the sequence are usually written with commas between them, the series is represented by a plus sign.

For example, the sequence given in the exercise \( c, c+d, c+2d, c+3d, ... \) is an infinite list when not restricted by the number of terms. However, when we talk about summing the first 'n' terms of this sequence, we refer to it as a series.

Understanding the difference between a sequence (the list itself) and a series (the sum of the list's terms) can greatly clarify the process of working with summation and can be central to solving such problems in mathematics.

Summation of Series

Summation refers to the process of adding the terms of a series together to find their total. In mathematics, and more specifically in the study of sequences and series, finding the sum of an arithmetic series is one of the fundamental concepts.

The formula to find the sum of the first n terms of an arithmetic series is \( S_n = \frac{n(a+l)}{2} \), where 'n' is the number of terms to be added, 'a' is the first term, and 'l' is the last term. This formula is derived by pairing terms from either end of the series that always sum to a constant value.

Here are the steps simplified:

• Identify the first term (a) and the last term (l)
• Determine the number of terms (n)
• Apply the formula for the sum of an arithmetic series

Applying this process to the exercise provided, we used the values of 'c' for the first term and \( c + (n - 1) * d \) for the last term, resulting in the derived formula for the series sum. This illustrates how summation is a powerful tool for understanding and computing the total of evenly increasing or decreasing sequences in both theoretical and practical scenarios.