Problem 51 - College Algebra

Question

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. $$ a_{n}=n+5 $$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of 1.

Step 1: Identify The Type of Sequence

To identify the type of sequence, locate the value of the first few terms. Calculate \(a_{1}=1+5\), \(a_{2}=2+5\), \(a_{3}=3+5\). This gives \(a_{1}=6\), \(a_{2}=7\), \(a_{3}=8\) respectively. If the difference between consecutive terms is a constant, the sequence is arithmetic.

Step 2: Calculate the Difference Between Terms

Calculate the difference between \(a_{2}\) and \(a_{1}\), and then \(a_{3}\) and \(a_{2}\). If \(a_{2}-a_{1} = a_{3}-a_{2}\), then the sequence is arithmetic. The calculated differences are: \(a_{2}-a_{1}=7-6=1\) and \(a_{3}-a_{2}=8-7=1\) As the difference 1 is the same, confirm the sequence is arithmetic.

Step 3: Find the Common Difference of the Arithmetic Sequence

Having confirmed that the sequence is arithmetic, identify the common difference. The common difference in an arithmetic sequence is the difference between any term and its preceding term. The difference identified in Step 2, which is 1, is the common difference.

Key Concepts

Common Difference

Understanding the 'common difference' is crucial when dealing with arithmetic sequences. An arithmetic sequence is a list of numbers with a specific pattern: each term is produced by adding a constant value to the previous term. This constant value is what we refer to as the common difference, often denoted by 'd'.

For instance, take the sequence of numbers 2, 4, 6, 8, and so on. This is an arithmetic sequence because every term after the first is created by adding 2; hence, the common difference is 2. To confirm that a sequence is arithmetic, you need to check that the difference between consecutive terms remains constant throughout the sequence. If you find disparities in these differences, then the sequence is not arithmetic. This elementary concept allows students to quickly identify and work with arithmetic sequences efficiently.

Sequence Determination

Determining the type of sequence you're dealing with is pivotal in understanding how to proceed with calculations and predictions within the sequence. In arithmetic sequences, you look for a common difference, but for geometric sequences, you seek a common ratio — the factor by which we multiply one term to get the next. When given a sequence, the first step is calculating the first few terms to look for a pattern.

In terms of progression, you can think of sequence determination like a detective's work: first gather the evidence (calculate terms), then look for patterns (are they increasing by addition or multiplication?), and finally draw a conclusion (is the sequence arithmetic, geometric, or neither?). Through effective sequence determination, you can unlock the secrets of the sequence and manipulate it to find any term within the progression.

Algebraic Sequences

Algebraic sequences are at the heart of understanding patterns and relationships in algebra. These sequences are represented by formulas that define the 'nth' term of a sequence as a function of 'n'. They can be arithmetic, geometric, or neither, appearing in various forms that encapsulate a rule for finding any term in the sequence.

An arithmetic sequence, like the one given in the example with the general term \( a_n=n+5 \), showcases the direct relationship between 'n' and each term of the sequence. The 'n' represents the position of the term in the sequence, while the formula gives the value of the term. Delving into algebraic sequences fosters the ability to generalize patterns, which is a foundational skill in mathematical reasoning and problem-solving. Remember, the key to mastering algebraic sequences is not just to perform calculations, but to grasp the 'why' and 'how' of the formation of such sequences.