Question

Multiple Choice Find the next term of the sequence \(0,3,9,18, \ldots\) (Lesson 1-1) (A) 21 (B) 30 (C) 54 (D) 162

Short answer

The next term is 30 (choice B).

Step-by-step solution

Examine the Sequence

The given sequence is \(0, 3, 9, 18, \ldots\). At first, observe how each term relates to the previous ones to detect any pattern. Let's focus on the differences between consecutive terms.

Find the Pattern in Differences

Calculate the differences between consecutive terms: \(3 - 0 = 3\), \(9 - 3 = 6\), \(18 - 9 = 9\).The differences are \(3, 6, 9, \ldots\), showing an arithmetic sequence with a constant difference of \(3\).

Determine the Pattern Rule

Noting that differences \(3, 6, 9, \ldots\) increase by \(3\) each time, the \(n^{th}\) difference can be represented as \(3n\), where \(n\) is the position of the difference in the sequence. We add these differences to the last term to find the next sequence term.

Calculate the Next Term

The next difference in the pattern is \(12\) (since the last difference was \(9\)). Add this to the last term in the sequence: \(18 + 12 = 30\).

Verify with Given Choices

The next term calculated is \(30\). Check the given choices:(A) 21, (B) 30, (C) 54, (D) 162. The correct answer corresponding to our calculation is (B) 30.

Key concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which each term after the first is obtained by adding a constant called the "common difference" to the preceding term. In the exercise given, the differences between consecutive terms are increasing, but they themselves form an arithmetic sequence:

• From 0 to 3 is an increase of 3.
• From 3 to 9 is an increase of 6.
• From 9 to 18 is an increase of 9.

These differences form the sequence: 3, 6, 9, ..., which increases by 3 each time.
Thus, the common difference in the differences is constant, further indicating the pattern in the changes between sequence terms.

Pattern Recognition

Pattern recognition in sequences involves identifying regularities or rules that dictate the formation of the sequence. In our exercise, we began by calculating the differences between each pair of successive terms. Recognizing these differences help us spot an underlying arithmetic pattern:

• The increasing set of differences (3, 6, 9...) suggests a consistent growth by 3.
• By noticing this, we could establish that the pattern forms an arithmetic sequence of its own.

This recognition of an arithmetic pattern within another sequence allows us to predict future values.

Mathematical Reasoning

Mathematical reasoning requires logical thought processes to extend sequences and conclude the next term. In our scenario, we reasoned that the differences between the original sequence values follow an arithmetic pattern. Using this understanding:

• We extend the pattern by calculating the next logical difference (12), which follows from continuing the sequence 3, 6, 9, ...
• Adding 12 to the last term, 18, gives us 30, determining the next term in the sequence.

Verification of this reasoning is confirmed by matching it with the choice provided in the multiple-choice options.
This step-by-step logical deduction ensures that our calculations are consistent with the pattern and helps confirm our understanding of sequences.