Problem 23 - Prealgebra

Question

Write an equation that describes each sequence. Then find the indicated term. \(14,15,16,17, \dots ; 16\) th term

Step-by-Step Solution

Answer

The 16th term is 29.

Step 1: Identify the Pattern

Examine the given sequence: 14, 15, 16, 17, ... The sequence increases by 1 with each subsequent term. This indicates a linear or arithmetic sequence where the common difference (d) is 1.

Step 2: Use the General Formula for Arithmetic Sequences

The general formula for the n-th term of an arithmetic sequence is \( a_n = a_1 + (n-1) imes d \), where \( a_1 \) is the first term and \( d \) is the common difference. For this sequence, \( a_1 = 14 \) and \( d = 1 \).

Step 3: Write the Equation for the n-th Term

Substitute the known values into the general formula: \( a_n = 14 + (n-1) \times 1 \). Simplify this to \( a_n = 14 + n - 1 = n + 13 \). This equation describes the n-th term of the sequence.

Step 4: Find the 16th Term

Now that we have the equation \( a_n = n + 13 \), find the 16th term by substituting \( n = 16 \). Calculate \( a_{16} = 16 + 13 = 29 \).

Key Concepts

Understanding Linear Sequences

A linear sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This steady increase or decrease makes it easy to predict future terms. Linear sequences are often referred to as arithmetic sequences.
They play a significant role in algebra and everyday life, ranging from calculating distances to predicting financial outcomes. In our example, the sequence is: 14, 15, 16, 17,...
Notice how each number is simply one more than the number before. This is a perfect case of a linear sequence because it follows a constant pattern by adding the same number each time.

Identifying the Common Difference

The common difference is a key aspect of an arithmetic sequence. It's the consistent amount that each term in the sequence changes, either by adding or subtracting it.
To find the common difference, subtract any term from the term that follows it. For example, in our sequence, subtract 14 from 15 to get 1. Likewise, subtract 15 from 16 and you still get 1.
This confirms that the change between terms is always 1, making this our common difference, denoted as \(d\). Understanding \(d\) helps identify how the sequence progresses.

The General Formula for the n-th Term

Creating an expression to find any term in an arithmetic sequence is crucial for efficient problem-solving. We use the general formula for the n-th term: \[a_n = a_1 + (n-1) \times d\] Key Variables:

• \(a_n\): the n-th term you want to find
• \(a_1\): the first term in the sequence
• \(n\): the position of the term
• \(d\): the common difference

For our sequence starting at 14 with \(d = 1\), the formula becomes: \[a_n = 14 + (n-1) \times 1\] Simplifying gives \(a_n = n + 13\). This formula can now be used to find any term without listing all numbers up to that point.

Applying the Formula for Arithmetic Sequences

In arithmetic sequences, applying the general formula allows for straightforward identification of any term. Let's say you need to find the 16th term of our example sequence.
Using the simplified formula \(a_n = n + 13\), substitute \(n = 16\) to find the term. This calculation looks like this: \[a_{16} = 16 + 13 = 29\] Thus, the 16th term is 29! With the formula established, you won't have to manually track each step of the sequence.
Simply plug in the desired term’s position, and you have your answer. Understanding and using this formula minimizes mistakes and speeds up the process.