Problem 51 - College Algebra

Question

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 \(a_{n}=n+5\) is an arithmetic sequence with a common difference of 1.

Step 1: Identify the sequence type

The sequence, \(a_{n}=n+5\), directly shows each term, \(a_{n}\), is the result of the ordinal number of the term, \(n\), plus a fixed number (5). This fits into the definition of an arithmetic sequence where there is a constant difference between the terms.

Step 2: Find the common difference

From the formula \(a_{n} = a_{1} + (n - 1) * d\) and the given \(a_{n} = n+5\), we can map \(a_{1}\) and \(d\) to the numbers in \(n+5\). Thus, \(a_{1}=6\) when \(n=1\) and the difference \(d\) can be derived by comparing the two formulas. Since we only have \(n\) in the equation with no multiplication or exponentiation, the difference \(d\) equals to 1.

Key Concepts

Common Difference

Understanding the common difference is crucial when studying arithmetic sequences. It is the consistent interval between consecutive terms in an arithmetic sequence. To visualize it, consider hopping on stepping stones across a river, where each stone is equidistant from the next. That regular distance is the sequence's common difference.

Referring to our exercise, the sequence was given as (a_n = n + 5). By observing the pattern of the sequence, we can determine the common difference. For instance, the first few terms are 6 (when n=1), 7 (when n=2), 8 (when n=3), and so on. Each number increases by 1. Therefore, unlike in other sequences where the difference might be difficult to decipher, here it is clear that the common difference ( d ) is 1. This consistent step demonstrates why this sequence is arithmetic in nature.

Sequence Patterns

Recognizing sequence patterns paves the way to mastering sequences in mathematics. A sequence pattern is like a rule that describes the relationship between successive terms. In other words, it's the secret code that, once cracked, unravels how numbers in the sequence are related to each other.

With algebraic sequences, like the one in our exercise (a_n = n + 5), the pattern can often be inferred directly from the algebraic expression. The formula indicates adding 5 to the current position number ( n) to get the sequence term. Hence, the number 5 consistently appears as part of the sum in each term of the sequence, and the change from one term to the next is always governed by the increment of 1 in n. This incremental change forms the backbone of an arithmetic sequence's pattern.

Algebraic Expressions

Delving into algebraic expressions opens the door to expressing mathematical ideas succinctly and powerfully. These expressions are combinations of numbers, variables, and operation symbols that represent a specific numerical relationship. The beauty of an algebraic expression is that it can condense a complex pattern into a compact formula that is universal for any number of terms in a sequence.

Take the sequence from our problem, (a_n = n + 5). This algebraic expression means that to find any term in the sequence, or a_n, you take the position of the term, symbolized by n, and add 5. By changing the value of n, you can generate every single term in the sequence. This expression is not only simple but represents an infinite list of standardized instructions to find any term’s value, demonstrating a powerful use of algebra in finding sequence patterns.