Problem 14 - Algebra 2

Question

Complete: 68 is the \(\underline{?}\) th term of the arithmetic sequence \(-2,3,8, \ldots\)

Step-by-Step Solution

Answer

68 is the 15th term of the sequence.

Step 1: Identify the First Term and Common Difference

The first term of the sequence is \(-2\). To find the common difference, subtract the first term from the second term: \(3 - (-2) = 5\). So, the common difference is \(5\).

Step 2: Set Up the Formula for the N-th Term

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

Step 3: Substitute the Given Values into the Formula

We know the n-th term is \(68\), the first term \(a\) is \(-2\), and the common difference \(d\) is \(5\). Substitute these values into the formula: \(68 = -2 + (n-1)\cdot5\).

Step 4: Solve the Equation for N

First, add \(2\) to both sides to isolate the term with \(n\): \[ 70 = (n-1)\cdot5 \]Divide both sides by \(5\) to solve for \(n-1\):\[ n-1 = 14 \]Add \(1\) to both sides to find \(n\):\[ n = 15 \]

Key Concepts

Common Difference

In an arithmetic sequence, the **common difference** is what makes the sequence unique.
It is a fixed value added to each term to get the next term. To find it, simply subtract the first term from the second term.
As an example, for the sequence \(-2, 3, 8, ...\), we calculate the common difference as follows:

• Second term: \(3\)
• First term: \(-2\)
• Common difference: \(3 - (-2) = 5\)

Recognizing this pattern helps identify how the sequence progresses and assists in predicting future terms effectively.
This steady addition plays an essential role in determining subsequent terms and understanding the general structure of arithmetic sequences.

N-th Term

The **n-th term** in an arithmetic sequence is expressed through a specific position in the sequence.
It allows us to identify any term's value based on its place without needing to list all the preceding terms.
This is calculated using the formula \(a_n = a + (n-1) \cdot d\), where:

• \(a_n\) is the n-th term
• \(a\) is the first term
• \(d\) is the common difference
• \(n\) is the term position

For instance, if given that \(68\) is our desired n-th term, the first term \(-2\), and the common difference \(5\):
1. Start with the equation: \(68 = -2 + (n-1) \cdot 5\)2. Rearrange and solve for \(n\).
This allows us to find which term position holds the value of \(68\) in the sequence.

Sequence Formula

The sequence formula provides a concise way to express the progression of an arithmetic sequence.
It's an efficient tool for determining any term's value in a sequence, incorporating the **first term** and the **common difference**.
The formula is written as \(a_n = a + (n-1) \cdot d\), and is pivotal in solving sequence-related problems.
By substituting known values into this formula, you can quickly compute unknown terms or positions.

• If you know the first term and the common difference, you can calculate all terms.
• If given a specific term value, the formula helps find its position \(n\) in the sequence.

Understanding this formula is essential for arithmetic sequences.
It provides clarity on how each term builds upon the last, illustrating the sequential nature of arithmetic calculations.