Problem 63 - Algebra 2

Question

Find the 27 th term of each sequence. $$ 5,8,11, \ldots $$

Step-by-Step Solution

Answer

The 27th term of the sequence is 83.

Step 1: Identify the Sequence

The given sequence is 5, 8, 11, ... which is an arithmetic sequence.

Step 2: Identify the Common Difference

Since this is an arithmetic sequence, there will be a common difference between consecutive terms. In this case, the difference is 8 - 5 = 3. So, the common difference \(d\) is 3.

Step 3: Identify the First Term

The first term in the sequence, represented as \(a_1\), is 5.

Step 4: Apply the Arithmetic Sequence Formula

To find any term in an arithmetic sequence, you can use the formula \(a_n = a_1 + (n - 1) \cdot d\). Here, \(n\) is 27, \(d\) is 3, and \(a_1\) is 5. Substitute these values in the formula to find the 27th term.

Step 5: Calculate the 27th Term

Substituting into the formula:\(a_{27} = 5 + (27 - 1) \cdot 3 = 5 + 26 \cdot 3 = 5 + 78 = 83\).

Key Concepts

Common Difference

The common difference in an arithmetic sequence is a key feature that characterizes the sequence itself. It is the value you add to each term to get to the next term in the sequence. It's important to identify this difference to work with arithmetic sequences efficiently. In our example sequence, which starts as 5, 8, 11,..., the common difference is calculated by subtracting the first term from the second term: 8 - 5 = 3. This means each term is 3 more than the previous term.
Finding the common difference helps to predict future terms and solve different problems related to arithmetic sequences.
Just remember:

• Identify any two successive terms.
• Subtract the earlier term from the later term to find the common difference.

This difference remains constant throughout the sequence. If the difference changes, then the sequence is not arithmetic.

Arithmetic Sequence Formula

The arithmetic sequence formula is a powerful tool for uncovering any term within the sequence. This formula is given by:
\[a_n = a_1 + (n - 1) \cdot d\]
Here, \(a_n\) is the \(n\)-th term you're trying to find. \(a_1\) is the first term of your sequence, \(d\) is the common difference, and \(n\) is the term number.

For our sequence, where \(a_1 = 5\), \(d = 3\), and \(n = 27\), we simply plug these values into the formula:

• \(a_{27} = 5 + (27 - 1) \times 3\)
• Solve inside the parentheses: \(26 \cdot 3\)
• Add 5 to it, and you have \(a_{27} = 83\)

With this formula, you don't need to list all the terms one by one. You can jump directly to any term you need in the sequence.

Sequence Terms

Terms in a sequence denote the individual elements in the arrangement of numbers. In an arithmetic sequence, each term is derived using the previous term plus the common difference.
A sequence like 5, 8, 11,... has its terms listed as 5 (first term), 8 (second term), 11 (third term), and so on.
Each term is calculated as follows:

• The first term is known and is denoted as \(a_1\).
• Every subsequent term can be derived by adding the common difference \(d\) to the previous term.

Knowing how to define each term allows for easy expansion of the sequence without explicitly outlining each one, especially useful when trying to find terms like the 27th without writing out every single one. Understanding this concept ensures effective computation of terms far down the line without difficulty.