Problem 50 - Algebra 2

Question

Use the given rule to write the \(4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},\) and 7 th terms of each sequence. $$ a_{n}=\frac{n+1}{n+2} $$

Step-by-Step Solution

Answer

The 4th, 5th, 6th, and 7th terms of the sequence are respectively \(\frac{5}{6}\), \(\frac{6}{7}\), \(\frac{7}{8}\) and \(\frac{8}{9}\).

Step 1: Calculate the 4th term

To find the 4th term, replace \(n\) by 4 in the given rule, i.e., \(a_4=\frac{4+1}{4+2}\).

Step 2: Simplify the expression

Simplify the expression to receive \(a_4=\frac{5}{6}\).

Step 3: Calculate the 5th term

To find the 5th term, replace \(n\) by 5 in the given rule, i.e., \(a_5=\frac{5+1}{5+2}\).

Step 4: Simplify the expression

Simplify the expression to receive \(a_5=\frac{6}{7}\).

Step 5: Calculate the 6th term

To find the 6th term, replace \(n\) by 6 in the given rule, i.e., \(a_6=\frac{6+1}{6+2}\).

Step 6: Simplify the expression

Simplify the expression to receive \(a_6=\frac{7}{8}\).

Step 7: Calculate the 7th term

To find the 7th term, replace \(n\) by 7 in the given rule, i.e., \(a_7=\frac{7+1}{7+2}\).

Step 8: Simplify the expression

Simplify the expression to receive \(a_7=\frac{8}{9}\).

Key Concepts

Understanding Term Calculation

Term calculation is a method used to find specific terms within a sequence. When a sequence is defined by a rule or formula, like the one given in the problem, we simply substitute the desired term number for the variable within the formula.

For instance, if the formula of a sequence is given as \( a_n = \frac{n+1}{n+2} \), finding the 4th term means replacing \( n \) with 4, resulting in \( a_4 = \frac{4+1}{4+2} = \frac{5}{6} \). This procedure is straightforward:

• Identify the formula that defines the sequence of terms.
• Substitute the desired term position (e.g., 4th, 5th) for the variable \( n \).
• Simplify the resulting expression to find the value of that specific term.

This approach applies to sequences in general, whether arithmetic, geometric, or defined by any other particular rule.

Role of Algebra in Sequences

Algebra is pivotal when working with sequences, especially when calculating specific terms or analyzing patterns. It provides a powerful toolset for manipulating and simplifying expressions.

In our exercise, the sequence formula \( a_n = \frac{n+1}{n+2} \) is expressed in an algebraic form. To compute specific terms, we rely on basic algebraic operations like addition and division.

• Addition helps in modifying the value of \( n \) within the formula, a crucial first step in pinpointing any term in the sequence.
• Division is used here to simplify the outcome, yielding the term's precise value.

Algebra also allows us to recognize patterns and deduce general properties of sequences, enhancing our ability to predict subsequent terms without direct calculation.

What Defines an Arithmetic Sequence?

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. However, not every sequence is arithmetic, and our exercise demonstrates such a case.

While the given sequence follows a clear pattern, it doesn't exhibit a constant difference typical of arithmetic sequences.

• In arithmetic sequences, this difference (called the common difference) remains unchanged as you progress from one term to the next.
• For instance, in the sequence 3, 7, 11, 15, each term increases by 4, representing a constant addition. Thus, the sequence is arithmetic with a common difference of 4.

Recognizing whether a sequence is arithmetic is crucial as it impacts how we compute terms and predict future values, contrasting with sequences defined by other types of rules.