Problem 40 - Algebra 2

Question

Write an equation for the nth term of each arithmetic sequence. \(-4,1,6,11, \ldots\)

Step-by-Step Solution

Answer

The nth term of the sequence is given by \(a_n = 5n - 9\).

Step 1: Identify the First Term

The first term of the arithmetic sequence is the very first number in the sequence. Here, the first term \(a_1\) is \(-4\).

Step 2: Determine the Common Difference

The common difference \(d\) of the arithmetic sequence is found by subtracting the first term from the second term. Thus, \(d = 1 - (-4) = 5\).

Step 3: Use the Formula for the nth Term

The formula to find the nth term \(a_n\) of an arithmetic sequence is given by \(a_n = a_1 + (n-1) \times d\).

Step 4: Plug Values into the Formula

Substitute \(a_1 = -4\) and \(d = 5\) into the nth term formula: \(a_n = -4 + (n-1) \times 5\).

Step 5: Simplify the Formula

Simplify the expression: \[a_n = -4 + 5(n-1) = -4 + 5n - 5 = 5n - 9\]. Thus, the nth term formula is \(a_n = 5n - 9\).

Key Concepts

nth term formula

The nth term formula is a powerful tool used in arithmetic sequences. It allows us to find any term in the sequence without needing to list out all the preceding numbers. This formula is especially useful when dealing with large sequences.
This formula is written as \( a_n = a_1 + (n-1) imes d \), where:

• \( a_n \) represents the nth term you want to find.
• \( a_1 \) is the first term of the sequence.
• \( d \) is the common difference between the terms.
• \( n \) is the position of the term in the sequence.

With this formula, you can simply plug in the known values and calculate the desired term. In our example, we simplified it to \( a_n = 5n - 9 \), making it straightforward to find any term.

common difference

In an arithmetic sequence, the common difference is the constant amount added to each term to get to the next one. It is denoted by \( d \), and its determination is key to understanding the sequence's progression.
To find the common difference:

• Subtract the first term \( a_1 \) from the second term \( a_2 \).

In our sequence, starting with \(-4\) and moving to \(1\), the common difference \( d \) is \( 1 - (-4) = 5 \). This tells us that each term increases by 5. By knowing this, you can predict future numbers in the sequence.

sequence pattern

The pattern of an arithmetic sequence follows a predictable and regular interval as specified by the common difference. Recognizing this pattern is essential for identifying and continuing the sequence.
The sequence starts with a first term, and each subsequent term is obtained by adding the common difference to the previous term.

• First Term: \(-4\)
• Second Term: \(-4 + 5 = 1\)
• Third Term: \(1 + 5 = 6\)
• Fourth Term: \(6 + 5 = 11\)

Once the pattern is clear, it's much simpler to calculate terms further along in the sequence. Sequences like these can often be continued infinitely, proceeding by the same constant additions.

arithmetics

Arithmetic sequences are a foundational concept in mathematics that focus on sequential numbers. They illustrate how numbers change through simple addition operations, influenced by a constant known as the common difference.
Understanding the basic operations of arithmetic provides insight into more complex mathematical ideas. This sequence type emphasizes how patterns and equations are formed by linear change, making computations easier and logical over numerous terms.

• Consists only of addition or subtraction.
• Brings predictability to number progressions.
• Lays groundwork for algebra and advanced math operations.

Using arithmetic principles, we can solve various real-life problems and better digest mathematical formulas.