Question

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{s_{n}\right\\}=\left\\{4^{n}\right\\} $$

Short answer

The sequence is geometric with a common ratio of 4. The first four terms are 4, 16, 64, and 256.

Step-by-step solution

Identify the General Term

Given the sequence \(\{s_{n}\} = \{4^{n}\}\), identify the general term. Here, the general term is \(a_{n} = 4^{n}\).

Verify the Geometric Nature

A sequence is geometric if there is a constant ratio between consecutive terms. To verify, calculate the ratio \(r\) by dividing the second term by the first term: \(r = \frac{a_{2}}{a_{1}} = \frac{4^{2}}{4^{1}} = \frac{16}{4} = 4\).

Find the Common Ratio

Since the ratio between consecutive terms \(r\) is constant and equal to 4, the sequence is confirmed to be geometric. Thus, the common ratio \(r\) is 4.

Calculate the First Four Terms

Use the general term \(a_n = 4^n\) to find the first four terms of the sequence: \(a_{1} = 4^{1} = 4\), \(a_{2} = 4^{2} = 16\), \(a_{3} = 4^{3} = 64\), and \(a_{4} = 4^{4} = 256\).

Key concepts

Common Ratio

In a geometric sequence, the common ratio is the factor by which we multiply one term to get the next term. It’s constant throughout the sequence. For our exercise, we found the common ratio by dividing the second term by the first term. For instance, given the sequence \{\{4^{n}\}\}, the second term is 16 (since 4^2), and the first term is 4 (since 4^1). Therefore, the common ratio, denoted as 'r,' is calculated as follows: \[ r = \frac{16}{4} = 4 \] This means each term in the sequence is 4 times the previous term. Understanding the common ratio helps us identify the nature of the sequence.

General Term

The general term of a sequence, often represented as \(a_n\), allows us to find any term in the sequence without listing all previous terms. In a geometric sequence, the general term can be described using the formula: \[ a_n = a_1 \times r^{(n-1)} \] Here, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term in the sequence. For our specific sequence \(\{4^n\}\), the first term \(a_1\) is 4, and the common ratio \(r\) is also 4. So, any term \(a_n\) can be written as: \[ a_n = 4^n \] This simplified form directly represents the value of the sequence at position \(n\).

Sequences and Series

Sequences and series are fundamental concepts in mathematics. A sequence is a list of numbers arranged in a specific order. There are different types of sequences such as arithmetic sequences, where the difference between consecutive terms is constant, and geometric sequences, where the ratio between consecutive terms is constant. A series, on the other hand, is the sum of the terms of a sequence. For example, for the sequence \(4, 16, 64, 256, ...\), the series would be the sum of these terms: \[ 4 + 16 + 64 + 256 + ... \] Understanding sequences helps us predict patterns and solve problems involving ordered sets of numbers.

Geometric Progression

A geometric progression (or geometric sequence) is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For the sequence given in the exercise, \(\{4^n\}\), we can clearly see that each term is multiplied by 4 to get the next term. For instance:

• The first term is 4 (which is \(4^1\))
• The second term is 16 (which is \(4^2\))
• The third term is 64 (which is \(4^3\))
• The fourth term is 256 (which is \(4^4\))

Hence, our sequence fits the definition of a geometric progression with a common ratio of 4. Identifying geometric progressions is crucial in various fields, including finance, computer science, and even nature.