Problem 81 - A Graphical Approach to College Algebra

Question

Suppose that \(a_{1}, a_{2}, a_{3}, \ldots\) and \(b_{1}, b_{2}, b_{3}, \ldots\) are arithmetic sequences. Let \(d_{n}=a_{n}+c \cdot b_{n},\) for any real number \(c\) and every positive integer \(n .\) Show that \(d_{1}, d_{2}, d_{3}, \ldots\) is an arithmetic sequence.

Step-by-Step Solution

Answer

Sequence \(d_n\) is arithmetic with first term \(d_1 = a_1 + c \cdot b_1\) and common difference \((d_a + c \cdot d_b)\).

Step 1: Identify the formulas for arithmetic sequences

In an arithmetic sequence, the difference between consecutive terms is constant. Let's denote the common differences as \(d_a\) for sequence \(a_n\) and \(d_b\) for sequence \(b_n\). Therefore, we can express the sequences as \(a_n = a_1 + (n-1)d_a\) and \(b_n = b_1 + (n-1)d_b\).

Step 2: Substitute Into the Formula for \(d_n\)

Given \(d_n = a_n + c \cdot b_n\), substitute the expressions for \(a_n\) and \(b_n\): \(d_n = \Big(a_1 + (n-1)d_a\Big) + c \cdot \Big(b_1 + (n-1)d_b\Big)\).

Step 3: Distribute and Simplify the Expression

Expand the expression for \(d_n\): \(d_n = a_1 + (n-1)d_a + c \cdot b_1 + c(n-1)d_b\). Combine like terms to get \(d_n = (a_1 + c \cdot b_1) + (n-1)(d_a + c \cdot d_b)\).

Step 4: Identify the Terms of the New Arithmetic Sequence

Recognize that \((a_1 + c \cdot b_1)\) acts as the first term, \(d_1\), and \((d_a + c \cdot d_b)\) is the common difference for this new sequence. Therefore, \(d_1, d_2, d_3, \ldots\) forms an arithmetic sequence with first term \(d_1 = a_1 + c \cdot b_1\) and common difference \(d_a + c \cdot d_b\).

Step 5: Conclusion: Confirm the Arithmetic Nature of \(d_n\)

Since \(d_n = (a_1 + c \cdot b_1) + (n-1)(d_a + c \cdot d_b)\), \(d_1, d_2, d_3, \ldots\) forms an arithmetic sequence, which completes the proof.

Key Concepts

Algebra

Algebra is an essential branch of mathematics that deals with symbols and the rules for manipulating these symbols. It involves everything from solving simple equations to developing complex models for real-world situations. In this exercise, algebra is used to represent arithmetic sequences and manipulate their terms. These sequences can be expressed with specific formulas, such as the nonlinear expressions given by \(a_n = a_1 + (n-1)d_a\) for one sequence and \(b_n = b_1 + (n-1)d_b\) for another. Here, algebra allows us to define the sequences mathematically and understand how changes in variables affect the overall sequence.In algebra, one often replaces numbers with symbols to better see patterns and solve problems. It helps us simplify expressions and solve for unknowns in equations. Using algebra in this context highlights its role in transforming complex verbal problems into solvable mathematical ones.

Sequence Formulas

To work with arithmetic sequences, it’s crucial to understand sequence formulas. These formulas express the nth term of a sequence based on its characteristics. An arithmetic sequence formula is given as \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.These formulas are powerful because they provide a quick way to find any term in the sequence without having to list all preceding terms. This saves time and reduces errors, especially for sequences with a large number of terms.In the given exercise, sequence formulas are used for expressing each sequence \(a_n\) and \(b_n\). By writing each sequence in terms of its first term and difference, we can manipulate them algebraically to form the sequence \(d_n\). This insight ultimately proves \(d_1, d_2, d_3, \ldots\) is also arithmetic, demonstrating the utility of sequence formulas in broader applications.

Common Difference

The concept of common difference is central to arithmetic sequences. It refers to the constant amount that each term increases or decreases by as you progress through the sequence. In algebraic terms, if \(d\) is the common difference, then you can express any term of an arithmetic sequence as \(a_n = a_1 + (n-1)d\).The common difference is significant because it dictates the direction and magnitude of the sequence. If \(d\) is positive, the sequence grows larger. If \(d\) is negative, it decreases. Understanding the role of the common difference allows for predictions about future terms and enables identification of patterns within the sequence.In the exercise, the idea of a common difference helps us understand why \(d_n = (a_1 + c \cdot b_1) + (n-1)(d_a + c \cdot d_b)\) leads to \(d_n\) forming an arithmetic sequence, with its own common difference of \(d_a + c \cdot d_b\). This shows how combined sequences maintain the characteristic pattern of arithmetic sequences when constructed correctly.