Problem 21 - Algebra 2

Question

Find the missing term of each arithmetic sequence. \(-16,\) _\(, 1, \ldots\)

Step-by-Step Solution

Answer

The missing term in the sequence is -7.5.

Step 1: Finding the Common Difference

In an arithmetic sequence, the common difference \(d\) is the same between all consecutive terms. Since we are given the first and third terms, we can calculate the common difference using the formula: \(d = \frac {T_{n} - T_{n-2}}{2}\), where \(T_{n}\) is the nth term of the sequence. For this sequence, \(T_{n}\) = 1 (third term), and \(T_{n-2}\) = -16 (first term). Therefore, \(d = \frac{1-(-16)}{2} = \frac{17}{2} = 8.5\).

Step 2: Calculating the Missing Term

Now, to find the missing term, we can use the formula for an arithmetic sequence: \(T_{n} = T_{n-1} + d\). The missing term is the second term, so we can replace \(T_{n-1}\) with -16 (the first term) , and 'd' with 8.5 (the common difference we calculated in the previous step). This gives us: \(T_{2} = (-16) + 8.5 = -7.5\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the value that separates consecutive terms. It remains constant throughout the entire sequence.
When you know at least two terms of a sequence, you can find this common difference. In our example, we were given the first term (-16) and the third term (1).
To find the common difference, we use the formula: \[d = \frac{T_n - T_{n-2}}{2}\]Here,

• \(T_n\) is the nth term of the sequence, and
• \(T_{n-2}\) is the term two places before \(T_n\).

This formula emphasizes that the difference between two non-consecutive terms is spread over multiple steps. In the exercise, we found the common difference \(d\) to be \(8.5\).
Understanding how to find the common difference is crucial, as it's necessary for calculating any term within an arithmetic sequence.

Missing Term

Finding a missing term in an arithmetic sequence involves using the common difference. In our sequence, we sought to find the second term, placed between (-16) and 1.
Once we calculated the common difference to be 8.5, we could use it to discover the missing term. By applying the formula, \[T_{n} = T_{n-1} + d\]we calculated the missing term. Here,

• \(T_{n-1}\) stands for the term directly before the missing term, which is the first term (-16), and
• \(d\), the common difference, is 8.5.

Thus, the missing second term was found by adding the common difference to the first term: \(T_2 = -16 + 8.5 = -7.5\).
Being able to calculate missing terms is valuable, as sequences often leave gaps that need to be filled for complete understanding.

Sequence Formula

The sequence formula for arithmetic sequences provides a straightforward way to plug in known values and solve for unknown ones. An arithmetic sequence follows the pattern where each term after the first is the sum of the previous term and the common difference.
The general formula for finding any term \(T_n\) is: \[T_n = T_1 + (n-1) \times d\]where:

• \(T_n\) represents the \(n\)th term,
• \(T_1\) is the first term of the sequence, and
• \(d\) is the common difference.

Using this formula, if you know the common difference and the first term, you can figure out any term's position without directly counting each step. This formula streamlines the process, making it easier to look ahead in sequences and solve more complex problems. Understanding and utilizing the sequence formula is key to mastering arithmetic sequences.