Problem 13 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Determine whether these terms can be the terms of a geometric sequence. If the sequence is geometric, find the common ratio. $$3,6,12,24, \dots$$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of 2.

Step 1: Definition of a Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted as \(r\).

Step 2: Determine Ratios Between Consecutive Terms

To check whether the sequence \(3, 6, 12, 24, \dots\) is geometric, compute the ratio between consecutive terms: \(\frac{6}{3} = 2\), \(\frac{12}{6} = 2\), and \(\frac{24}{12} = 2\).

Step 3: Conclusion of Common Ratio

Since the ratio between each pair of consecutive terms is the same (\(r = 2\)), the sequence is indeed geometric.

Step 4: Identify the Common Ratio

The common ratio \(r\) of this geometric sequence is \(2\). Each term is obtained by multiplying the previous term by \(2\).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is pivotal. It is the fixed number that you multiply with one term to get the next term in the sequence. This is essentially the heart of how a geometric sequence is formed. For the sequence to be geometric, this ratio must be consistent across all pairs of consecutive terms.

For example, in the sequence given: 3, 6, 12, 24, we calculated the common ratio by dividing each term by its preceding term. Thus,

• \(\frac{6}{3} = 2\)
• \(\frac{12}{6} = 2\)
• \(\frac{24}{12} = 2\)

As you can see, the ratio remains constant, making the common ratio \(2\), and therefore, the sequence geometric.

Whenever you encounter a sequence you suspect might be geometric, checking the common ratio is your go-to strategy. It is this consistent multiplication factor that verifies the sequence's geometric nature.

Sequence of Numbers in Geometric Progression

A geometric sequence is simply a sequence of numbers where each term after the first is determined by multiplying the previous one by a constant number. This type of sequence is different from an arithmetic sequence, where you add a constant value. Instead, you're repeatedly multiplying by the common ratio.

In the sequence we've been exploring (3, 6, 12, 24), each number multiplies to the next by \(2\).

• Start with 3
• Multiply by 2 to get the next term: 6
• Multiply by 2 again: 12
• Once more by 2: 24

This simple multiplication process defines the geometric nature of the sequence, where every step lies in the power of its common ratio.

By understanding how each number follows the previous in this chain of multiplication, you grasp the essence of a geometric sequence.

Multiplying Terms in a Geometric Sequence

When dealing with a geometric sequence, multiplying terms is your core operation. Once you've identified the common ratio, you can generate any term in the sequence by multiplying the preceding term by this ratio.

For the sequence 3, 6, 12, 24, we've already determined that the common ratio is 2.

• Starting from the first term, 3: multiply by 2 to get 6
• From 6: multiply by 2 to get 12
• From 12: multiply by 2 to reach 24

Each term 'sets the stage' for creating the next, showcasing how essential the act of multiplication is within a geometric sequence. This principle is incredibly useful when asked to find subsequent terms without listing all of them. Simply keep multiplying by \(r\), and you'll always stay on track.

Understanding this multiplication process allows you to extend the sequence infinitely or find any specific term based on its position.