Question

Describe the pattern in each sequence, and use the pattern to find the next three terms. \(\Delta, \infty, \Delta, \Delta, \infty, \Delta, \Delta, \Delta, \infty\) __, __, __

Short answer

\( \triangle, \triangle, \ \triangle\)

Step-by-step solution

Identify the Sequence Pattern

Examine the given sequence: \(\triangle, \infty, \triangle, \triangle, \infty, \triangle, \triangle, \triangle, \infty\). Notice that the term \( \infty\) appears after a certain number of \( \triangle\) symbols. Count the number of \( \triangle\) symbols between each \( \infty\): 1) \(\triangle, \infty\) - (1 \( \triangle\)) 2) \(\triangle, \triangle, \infty\) - (2 \( \triangle\)) 3) \(\triangle, \triangle, \triangle, \infty\) - (3 \( \triangle\)). Observe a pattern where the number of \( \triangle\) symbols increases by one before each \( \infty\).

Continue the Pattern

Based on the identified pattern, predict the next set of symbols. The pattern shows us that after three \( \triangle\) symbols comes an \( \infty\). To continue the sequence, we need to follow the increasing number of \( \triangle\) symbols: 4) The next set after the sequence ends with \( \infty\) should have four \( \triangle\) symbols.

Determine the Next Three Terms

Following the pattern: - After the three \( \triangle\) symbols that precede the last \( \infty\), the next term should be: \( \triangle\) - Continuing, we then have another \( \triangle\) - Finally, we have one more \( \triangle\), followed by the next symbol which should be \( \infty\). Thus, the next three terms in the sequence are: \( \triangle, \triangle, \ \triangle\).

Key concepts

Pattern Recognition

Identifying patterns is crucial in mathematics, especially when dealing with sequences. In this exercise, we examine the sequence: \( \Delta, \infty, \Delta, \Delta, \infty, \Delta, \Delta, \Delta, \infty \). Recognizing patterns allows us to predict future terms and understand the behavior of sequences.
Let's break it down:

The symbol \(\infty \) follows \(\Delta \) symbols.

The number of \(\Delta \) symbols increases by one each time before an \(\infty \) symbol appears.

By identifying this pattern, we can predict the next terms accurately. First, determine how often \(\infty \) appears, then check the increasing number of \(\Delta \) symbols in between.
This logical sequence helps us spot the next terms easily.

Mathematical Sequences

A mathematical sequence is a list of numbers or symbols arranged in a specific order. In this case, our sequence is a mix of two symbols: \( \Delta \) and \( \infty \). Understanding sequences involves recognizing the order and progression of terms.
Here, the progression relates to the number of \( \Delta \) symbols between each \( \infty \) symbol, which increases by one each time.
Let's break down the current sequence:

• \( \Delta, \infty \) \ (1 \( \Delta \))
• \( \Delta, \Delta, \infty \) \ (2 \( \Delta \))
• \( \Delta, \Delta, \Delta, \infty \) \ (3 \( \Delta \))

Following the pattern, the next set should have four \( \Delta \) symbols. Sequences help us look for such structures, predict new terms, and understand their relationships.

Algebra Concepts

In algebra, patterns and sequences often lead to forming algebraic expressions or functions. Though our sequence uses symbols, the idea is similar. The identified pattern shows an increasing number of \( \Delta \) symbols before each \( \infty \) symbol.
By understanding this, we can use algebraic thinking to predict the continuation:

• If \( n \) represents the number of \( \Delta \) symbols before the \( n \)th \( \infty \) symbol, then each \( \Delta \) count increases by one for each successive \( \infty \).
• For example, after three \( \Delta \), we get \( \infty \). The next set should have four \( \Delta \) before hitting \( \infty \).

Algebra helps us develop such patterns and understand their mathematical relationships. In essence, we use algebra to describe sequences in a general form, allowing deeper insights into their behavior and prediction of future terms.