Problem 88 - Algebra 2

Question

Which arithmetic sequence includes the term 27\(?\) I. \(a_{1}=7, a_{n}=a_{n-1}+5\) \(\quad\) II. \(a_{n}=3+(n-1) 4\) \(\quad\) III. \(a_{n}=57-6 n\) F. I only G. I and II only H. II and III only J. \(1,11,\) and 111

Step-by-Step Solution

Answer

J. All sequences (I, II, and III) include the term 27.

Step 1: Analysis of Sequence I

First, let's check if the term 27 is part of sequence I. The sequence is given by \(a_{n}=a_{n-1}+5\). The first term \(a_{1}\) is given as 7. An arithmetic sequence can also be represented as \(a_{n}=a_{1}+(n-1)d\), where \(d\) is the common difference. Here, \(d=5\). So, we substitute these values into the formula to get \(27=7+(n-1)5\).

Step 2: Solve for n for Sequence I

Solving for n in the equation \(27=7+(n-1)5\), we get \(n=5\). Since this is a positive integer, it confirms that the term 27 is part of sequence I.

Step 3: Analysis of Sequence II

Next, let's check Sequence II which is described by the formula \(a_{n}=3+(n-1)4\). Again, we substitute the term 27 into this formula giving us \(27=3+(n-1)4\).

Step 4: Solve for n for Sequence II

Solving the equation \(27=3+(n-1)4\), we get \(n=7\). This also results in a positive integer, which confirms that 27 is part of sequence II.

Step 5: Analysis of Sequence III

Lastly, we check Sequence III which is given as \(a_{n}=57-6n\). As usual, we substitute the term 27 and solve for n using the equation \(27=57-6n\).

Step 6: Solve for n for Sequence III

Solving the equation \(27=57-6n\) yields \(n=5\). This suggests that the term 27 is also part of the Sequence III.

Key Concepts

Common Difference

The common difference in an arithmetic sequence is a key characteristic that sets each term apart from the next by a constant value. This consistent difference between consecutive terms is what defines and identifies the sequence as "arithmetic."

• In Sequence I, the common difference is 5. This means every term after the first is 5 more than the previous term.
• For Sequence II, the common difference is 4. Each term increases by 4 over the term before it.
• In Sequence III, although formatted slightly differently, the common difference can be understood as -6, because each term decreases by 6.

Understanding the common difference allows us to predict any term by repeatedly adding it to the starting number. It is fundamental in sequence analysis, as it enables us to discern patterns and solve problems regarding term identification.

Sequence Analysis

Sequence analysis involves looking at the structure and pattern within a sequence to understand its characteristics and behavior. For arithmetic sequences, it's crucial to identify the order and difference in its sequence of numbers. To thoroughly analyze a sequence:

• Identify the starting term ( a_1 ).
• Determine the common difference ( d ).
• Use these to investigate whether a particular term exists within the sequence.

For example, by examining the given sequences:
- Sequence I starts at 7 and shows an increase of 5 each step;
- Sequence II begins at 3 with each term increasing by 4;
- Sequence III is more unique but still follows a constant pattern, decreasing by 6 each term. This analysis helps establish whether certain terms, such as 27, exist within the sequences by solving the ordering using these patterns.

Term Identification

Term identification in arithmetic sequences involves determining whether a particular number is part of the sequence. This process requires substitution of the particular term into the given formula and solving for its position or ordinal number in the sequence.For instance, in the exercise:

• For Sequence I, we need to find if 27 can be expressed as an element of the sequence following the pattern: \[ 27 = 7 + (n-1)5 \]
• Similarly, for Sequence II: \[ 27 = 3 + (n-1)4 \]
• And for Sequence III: \[ 27 = 57 - 6n \]

By solving these equations for n, we can ascertain whether 27 corresponds to a valid position in each sequence. If n results in a positive integer, then 27 is indeed a term in that sequence.

Arithmetic Sequence Formula

The arithmetic sequence formula is an equation that helps to determine any term in an arithmetic sequence. The standard formula is:\[ a_{n} = a_{1} + (n-1) imes d \]Where:

• \( a_{n} \) is the nth term you want to find.
• \( a_{1} \) is the first term of the sequence.
• \( n \) is the term's position within the overall sequence.
• \( d \) is known as the common difference.

Using this formula, one can find any term's value within a sequence, provided the first term and the common difference are known. It simplifies the process of identifying any significant values, confirming the presence of numbers like 27 in the sequences analyzed, by allowing straightforward calculation of the necessary sequence position.