Problem 41 - Algebra and Trigonometry

Question

Which term of the arithmetic sequence \(1,4,7, \ldots\) is \(88 ?\)

Step-by-Step Solution

Answer

The 30th term of the sequence is 88.

Step 1: Identify the sequence parameters

An arithmetic sequence is defined by its first term (\(a_1\)) and common difference (\(d\)). From the given sequence \(1, 4, 7, \ldots\), we can see that \(a_1 = 1\) and the common difference \(d = 4 - 1 = 3\).

Step 2: Write the formula for the \(n\)-th term

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

Step 3: Set up the equation for the \(n\)-th term

We want to find which term \(a_n\) equals \(88\). So, we set up the equation: \(88 = 1 + (n-1) \times 3\).

Step 4: Solve the equation for \(n\)

First, simplify the right side: \(88 = 1 + 3n - 3\). This leads to \(88 = 3n - 2\). Add 2 to both sides, resulting in \(90 = 3n\). Finally, divide both sides by 3: \(n = \frac{90}{3} = 30\).

Step 5: Verify the solution

Check if the 30th term is indeed 88 by calculating it using the formula: \(a_{30} = 1 + (30-1) \times 3 = 1 + 87 = 88\). The calculation confirms our solution.

Key Concepts

Common Difference

An arithmetic sequence is defined by having equal differences between consecutive terms. This difference is termed the "common difference." It's a crucial component in identifying and working with arithmetic sequences. In our exercise, the sequence provided is \(1, 4, 7, \ldots\). To find the common difference \(d\), subtract the first term from the second term:

• Second term: 4
• First term: 1
• Common difference \(d = 4 - 1 = 3\)

The common difference \(d\) tells us how much each subsequent term increases by as you move from one element of the sequence to the next. Understanding the common difference not only helps navigate among the sequence but is also pivotal for calculating other terms with the provided formulas.
The sequence continues by adding this common difference to each subsequent term.

n-th Term Formula

The n-th term formula is the backbone of finding the value of any term in an arithmetic sequence. For an arithmetic sequence, the n-th term, denoted as \(a_n\), can be calculated using the initial term \(a_1\) and the common difference \(d\). The formula is:\[a_n = a_1 + (n-1) \times d\]This formula is derived from adding the common difference \(d\)

• \((n-1)\times\) times to the first term \(a_1\)

For our given sequence, \(a_1 = 1\) and \(d = 3\), the formula becomes:\[a_n = 1 + (n-1) \times 3\]This allows us to easily find any term in the sequence without listing all the preceding numbers. It’s like having a map that tells us exactly where any specific term is located, based on its position \(n\).

Sequence Parameters

Sequence parameters are crucial in defining the nature of any arithmetic sequence. The two primary parameters are:

• \(a_1\) - the first term of the sequence
• \(d\) - the common difference

These parameters determine how the sequence behaves and how each subsequent term is generated. For example, in our problem:

• The first term \(a_1 = 1\)
• The common difference \(d = 3\)

Armed with just these parameters, we can calculate any term using the n-th term formula, validating the progression of the sequence. Understanding these parameters is critical for generating sequences and solving problems related to them, such as our task to find which term in the sequence is 88.