Problem 29 - Prealgebra

Question

Write an equation that describes each sequence. Then find the indicated term. \(3,8,13,18, \dots ; 70\) th term

Step-by-Step Solution

Answer

The equation is \(a_n = 5n - 2\). The 70th term is 348.

Step 1: Identify the Type of Sequence

Observe the given sequence: \(3, 8, 13, 18, \dots\). Each term increases by a constant 5. This indicates that the sequence is an arithmetic sequence.

Step 2: Find the Common Difference

In an arithmetic sequence, the common difference \(d\) can be found by subtracting any term from the next term. Calculating it, we have \(d = 8 - 3 = 5\).

Step 3: Write the Equation for the nth Term

The formula for the nth term \(a_n\) of an arithmetic sequence is given by \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference. Plugging in the values, \(a_n = 3 + (n-1) \times 5\).

Step 4: Simplify the nth Term Equation

Simplify the equation: \(a_n = 3 + 5n - 5 = 5n - 2\). This is the equation that describes the sequence.

Step 5: Find the 70th Term

Substitute \(n = 70\) into the equation \(a_n = 5n - 2\) to find the 70th term. Calculate it: \(a_{70} = 5 \times 70 - 2 = 350 - 2\).

Step 6: Complete the Calculation

Finish the calculation: \(a_{70} = 348\). Thus, the 70th term is 348.

Key Concepts

Understanding Common Difference

In arithmetic sequences, the concept of a common difference is what defines the unique attribute of the sequence. The common difference, denoted as \(d\), is a constant value that each term is incremented by to get to the next term in the sequence. For any arithmetic sequence:

• Take any term and subtract the previous term from it.
• If the sequence is \(3, 8, 13, 18, \ldots\), you'd find \(d\) by calculating \(8 - 3 = 5\).

Each term of the sequence relies on \(d\) for its calculation. The consistent change across the sequence ensures it’s arithmetic, distinguishing it from other types of sequences.

The nth Term Formula Explained

The nth term formula is a powerful tool for determining any term in an arithmetic sequence without listing out all terms until you reach the desired one. This formula is written as:

• \(a_n = a_1 + (n-1)d\)
• Where \(a_n\) is the term you're trying to find, \(a_1\) is the first term, and \(n\) is the position of the term in the sequence.

For example, in our sequence, using \(a_1 = 3\), and \(d = 5\), you can easily substitute these values into the formula to find any term, such as the 70th term, without hassle. This formula is your go-to for efficiency in arithmetic sequence problems.

Building the Sequence Equation

The beauty of arithmetic sequences is encapsulated in their formulaic predictability. After finding both the first term and the common difference, you can write the sequence equation. When you have determined these elements, the sequence equation becomes:

• \(a_n = 5n - 2\)

You arrive at this by insertion and simplification of the general formula, converting it into a straightforward sequence equation. This equation embodies all terms in the sequence, underscoring both direction (through \(n\)) and change (through \(d\)). The sequence equation is your compass in navigating through an arithmetic sequence!

Grasping Prealgebra Concepts

Prealgebra primarily entails understanding foundational math concepts, which include recognizing patterns such as arithmetic sequences among numbers. It prepares you for algebra through concepts such as:

• The idea of using a consistent rule (like a common difference) to move through a series of numbers.
• Recognizing the need for formulas to find terms in an arithmetic sequence efficiently.

This exercise enriches your prealgebra foundation by applying the arithmetic sequence formula to solve for unknown terms. The prealgebra stage is essential in setting the ground for algebraic thinking and enhancing problem-solving skills.