Problem 46 - Prealgebra

Question

Write an equation that describes each sequence. Then find the indicated term. $$4,11,18,25, \dots ; 100 \text { th term }$$

Step-by-Step Solution

Answer

The 100th term of the sequence is 697.

Step 1: Identify the Sequence

The sequence given is an arithmetic sequence, which is shown by a constant difference between consecutive terms.

Step 2: Find the Common Difference

Subtract the first term from the second term: \( 11 - 4 = 7 \). The common difference \( d \) is 7.

Step 3: Write the General Formula

The 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. Here, \( a_1 = 4 \) and \( d = 7 \). Substitute to find the equation: \( a_n = 4 + (n-1) imes 7 \).

Step 4: Simplify the Equation

Distribute and simplify the equation: \( a_n = 4 + 7n - 7 = 7n - 3 \). This is the equation for the \( n \)-th term.

Step 5: Find the 100th Term

Substitute \( n = 100 \) into the equation \( a_n = 7n - 3 \): \( a_{100} = 7(100) - 3 = 700 - 3 = 697 \).

Step 6: Conclusion

The 100th term of the sequence is 697.

Key Concepts

Understanding the Sequence Formula

In arithmetic sequences, the sequence formula is paramount for capturing the essence of patterns found in numbers. Usually, these sequences form by adding a constant value to progress from one term to the next. When dealing with an arithmetic sequence, we are talking about numbers organized in a sequential order where the difference (called the common difference) between consecutive terms remains consistent. The sequence formula for the nth term is given by:

• \( a_n = a_1 + (n-1) \times d \)

Here, \( a_n \) represents the nth term. This formula is crucial as it allows you to find any term in the sequence without listing all previous terms. This makes calculation efficient and straightforward.

Discovering the Common Difference

The common difference in an arithmetic sequence is a defining characteristic. It tells you how much each term increases compared to the one before. To find the common difference \( d \), simply subtract any term from the one following it.
For example, in the sequence 4, 11, 18, 25,..., the common difference can be calculated as follows:

• \( d = 11 - 4 = 7 \)

Recognizing and calculating the common difference is crucial. It not only helps in constructing the sequence formula but also determines how quickly the terms in the sequence grow or decrease.

Finding the N-th Term

The notion of the n-th term is quite central to understanding any sequence. When asked to find the n-th term, it's like pointing to the exact room number in a long hallway—the position and the value at that position.
The formula for finding the n-th term in an arithmetic sequence is straightforward once the common difference and the first term are known.

• \( a_n = a_1 + (n-1) \times d \)

For our specific sequence, we found: \( a_n = 7n - 3 \).Plugging in a specific number for \( n \), such as 100, will provide us with the precise value of the sequence at that spot: \( a_{100} = 697 \). This efficient method sidesteps lengthy lists or repeated calculations.

Applying Mathematical Problem Solving Techniques

Mathematical problem solving involves several steps that bring clarity and understanding to complex problems. When tackling an arithmetic sequence, key steps ensure a smooth problem-solving path:

• Identify the type of sequence: Is it arithmetic? Check for constant differences.
• Calculate the common difference using the subtraction method.
• Write the general formula using known values such as the first term and the common difference.
• Apply algebraic manipulations to simplify the formula, thereby making it easy to find any term.
• Use substitution techniques for finding specific terms, like the 100th term.

These problem-solving strategies ensure that processes are not just completed but understood deeply, empowering you to tackle similar problems with confidence and precision.