Problem 71 - Algebra for College Students

Question

Before developing the formula \(a_{n}=a_{1}+(n-1) d\), we stated the equation \(a_{k+1}-a_{k}=d\). In your own words, explain what this equation says.

Step-by-Step Solution

Answer

The equation states that the difference between consecutive terms in an arithmetic sequence is constant, defining the sequence type.

Step 1: Understanding Notation

The given equation is written as \(a_{k+1} - a_{k} = d\). Here, \(a_{k+1}\) represents the \((k+1)^{th}\) term of an arithmetic sequence, \(a_{k}\) represents the \(k^{th}\) term, and \(d\) is the common difference of the sequence.

Step 2: Analyzing the Equation

The equation \(a_{k+1} - a_{k} = d\) suggests that the difference between two consecutive terms in an arithmetic sequence is a constant, \(d\). This means every term after the first term is formed by adding \(d\) to the previous term.

Step 3: Implication of the Equation

The equation implies that the sequence increases or decreases by a fixed amount, \(d\), as you move from one term to the next. This constant difference \(d\) is what defines the sequence as arithmetic.

Step 4: Connection to the Formula

The equation is a foundational idea leading to the general arithmetic sequence formula \(a_{n}=a_{1}+(n-1)d\), where \(a_{1}\) is the first term and \(n\) is the term number. The formula results from adding the common difference \(d\) repeatedly to the first term \(a_{1}\).

Key Concepts

Common Difference

In an arithmetic sequence, the term "common difference" is a key concept. It refers to the constant quantity added to each term to get the next term in the sequence. This is indicated by the variable \(d\) in the equation \(a_{k+1} - a_{k} = d\).
This equation tells us that the difference between any two consecutive terms is always the same, and it's precisely this consistent difference that makes the sequence arithmetic.

• If \(d\) is positive, each term is larger than the previous, leading to an increasing sequence.
• If \(d\) is negative, each term is smaller than the previous, which results in a decreasing sequence.

The common difference remains unchanged throughout the sequence, making it a critical aspect of understanding and working with arithmetic sequences.
Identifying the common difference allows you to easily predict future terms of the sequence.

Sequence Terms

Each term in an arithmetic sequence is created by continually adding the common difference \(d\) to the previous term.
Consider our expressions \(a_1, a_2, a_3,\ldots, a_n\).
Here, each term \(a_k\) can be described by the equation \(a_{k} = a_{1} + (k-1)d\).
This means:

• To find any term, you start with the first term \(a_1\).
• Add the common difference \(d\) enough times to reach the desired position \(n\).

For instance, if you want the 5th term, you're adding the common difference four times to the first term. This systematic approach explains why arithmetic sequences are also known as linear progressions, as each step or term is linearly dependent on its predecessor.

Linear Progression

An arithmetic sequence is an example of a linear progression, meaning it represents a series of numbers where each term is a linear function of its position. Let's break it down further:
The formula for arithmetic sequences, \(a_{n}=a_{1}+(n-1)d\), reveals its linear nature:

• The relationship between the sequence's position \(n\) and its value \(a_n\) is straightforward.
• Each term is constructed linearly, using simple addition and multiplication involving \(n\) and \(d\).

This linearity is what differentiates arithmetic sequences from other types like geometric sequences, where the terms multiply rather than add.
Because these progressions increase or decrease at a constant rate, they form a straight line when graphed on a coordinate system. This trait is why they're called linear progressions, highlighting the uniformity and predictability of arithmetic sequences.