Problem 30 - Algebra 2

Question

Find the missing term of each arithmetic sequence. \(\ldots 99, \square, 66, \dots\)

Step-by-Step Solution

Answer

The missing term of the arithmetic sequence is 82.5.

Step 1: Identify the given terms and their positions

In an arithmetic sequence, any term is equal to the previous term plus the common difference. The given terms are the third term \(a_3 = 99\) and the fifth term \(a_5 = 66\). Our goal is to find the fourth term, \(a_4\).

Step 2: Find the common difference

The formula to find any term in an arithmetic sequence is: \(a_n = a_1 + (n-1)d\). However, the first term is not given in this problem. Instead, let's use the given third and fifth terms to find the common difference (d): \((a_5 - a_3) / (5 - 3) = (66 - 99) / 2 = -16.5\).

Step 3: Calculate the missing term

Since the common difference is -16.5, we can find the fourth term (the missing term) by adding the common difference to the third term: \(a_4 = a_3 + d = 99 - 16.5 = 82.5\)

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is crucial. It is the constant number you add or subtract from one term to the next. This number helps to define the sequence.

• The common difference is found by subtracting any term in the sequence from the subsequent term.
• In our exercise, to get from 99 to 66 (two terms apart), we calculate: \[(66 - 99) / (5 - 3) = -16.5\]
• The common difference here is -16.5, meaning each term is 16.5 less than the one before it.

Understanding the common difference allows us to construct or extend the sequence effectively. Without this value, figuring out future or missing terms would be much more complicated.

Missing Term

Finding the missing term in an arithmetic sequence means identifying the unknown number positioned between known terms.

• First, identify the given term positions and values. For example, here, we know the 3rd term (99) and the 5th term (66).
• Use the common difference \(-16.5\) to find the missing term.
• We need to calculate the 4th term by adding the common difference to the 3rd term:\[a_4 = 99 + (-16.5) = 82.5\]

By the power of arithmetic sequences, you've found that the 4th term is 82.5! Recognizing such patterns makes arithmetic sequences easier to grasp, allowing missing terms to be identified swiftly.

Sequence Calculation

Arithmetic sequences follow a straightforward formula which allows for easy calculation of terms:

• The general formula is: \[a_n = a_1 + (n-1) \, d\]where:
  • \(a_n\) is the term you are trying to find.
  • \(a_1\) is the first term.
  • \(n\) is the term position.
  • \(d\) is the common difference.
• If the first term is not given, use any two known terms. For instance, given 99 and 66, calculate \(d\) to uncover missing pieces:
• Once the common difference is known, substitute it to find any unknown terms by continuing the sequence.

This procedure can be applied to any arithmetic sequence to find unknown terms either before, between, or after the given terms.