Question

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{c_{n}\right\\}=\left\\{\frac{2^{n-1}}{4}\right\\} $$

Short answer

Common ratio is 2. First four terms are \( \frac{1}{4}, \frac{1}{2}, 1, 2 \).

Step-by-step solution

Identify the General Term

The given sequence is defined by the general term: \( c_{n} = \frac{2^{n-1}}{4} \)

Write Out the First Four Terms

To check if the sequence is geometric, write out the first four terms by substituting \(n = 1, 2, 3, 4\): \( c_{1} = \frac{2^{1-1}}{4} = \frac{2^0}{4} = \frac{1}{4} \)\( c_{2} = \frac{2^{2-1}}{4} = \frac{2^1}{4} = \frac{2}{4} = \frac{1}{2} \)\( c_{3} = \frac{2^{3-1}}{4} = \frac{2^2}{4} = \frac{4}{4} = 1 \)\( c_{4} = \frac{2^{4-1}}{4} = \frac{2^3}{4} = \frac{8}{4} = 2 \)

Verify the Common Ratio

Verify that the sequence is geometric by finding the ratio between successive terms: \( \frac{c_{2}}{c_{1}} = \frac{\frac{1}{2}}{\frac{1}{4}} = 2 \)\( \frac{c_{3}}{c_{2}} = \frac{1}{\frac{1}{2}} = 2 \)\( \frac{c_{4}}{c_{3}} = \frac{2}{1} = 2 \) Since the ratio is constant, the sequence is geometric.

State the Common Ratio and First Four Terms

The common ratio \( r \) is 2. The first four terms of the sequence are \( \frac{1}{4}, \frac{1}{2}, 1, \) and \(2\).

Key concepts

Common Ratio

In a geometric sequence, each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
This ratio is a key feature that defines a geometric sequence.
The common ratio can be calculated by dividing any term by the previous term. For the sequence defined by the general term \( c_{n} = \frac{2^{n-1}}{4} \), we can see that the ratio between each pair of successive terms is 2.

For instance, if you divide the second term by the first term, you get \( \frac{\frac{1}{2}}{\frac{1}{4}} = 2 \). Similarly, dividing the third term by the second term: \( \frac{1}{\frac{1}{2}} = 2 \), and so on.

• This constant ratio confirms that the sequence is geometric.

Sequence Terms

Each element in a sequence is known as a term.
In the given sequence, the terms can be calculated by substituting different values of n into the general term formula
\( c_{n} = \frac{2^{n-1}}{4} \).

To find the first four terms:

• For \(n=1\), the term is \( c_{1} = \frac{2^{1-1}}{4} = \frac{1}{4} \)
• For \(n=2\), the term is \( c_{2} = \frac{2^{2-1}}{4} = \frac{2}{4} = \frac{1}{2} \)
• For \(n=3\), the term is \( c_{3} = \frac{2^{3-1}}{4} = \frac{4}{4} = 1 \)
• For \(n=4\), the term is \( c_{4} = \frac{2^{4-1}}{4} = \frac{8}{4} = 2 \)

These values are the first four terms of the sequence: \(\frac{1}{4}\), \(\frac{1}{2}\), \( 1 \), and \( 2 \).

Geometric Progression

A geometric progression, or geometric sequence, is a sequence of numbers where each term after the first is the product of the previous term and a constant known as the common ratio.

The sequence \( c_{n} = \frac{2^{n-1}}{4} \) is a perfect example of a geometric sequence: each term is derived by multiplying the previous term by 2, our common ratio.

Unlike arithmetic sequences, where the difference between terms is constant, geometric sequences focus on the product relationship between terms, which can grow or shrink exponentially depending on the value of the common ratio.

Understanding geometric progressions is essential in various branch of mathematics, especially in algebra and real-life applications like population growth, interest rates, and computer algorithms.

Algebra

Algebra is the branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations.

It plays a crucial role in understanding sequences. To define and work with sequences like the geometric one given by \( c_{n} = \frac{2^{n-1}}{4} \), algebraic expressions and methods are essential.

Breaking down this particular sequence algebraically involves understanding exponents and fractions. For example, knowing that \( 2^{n-1} \) grows exponentially as \( n \) increases helps us see why the sequence terms change so dramatically.

Algebra also helps us calculate the common ratio and verify the nature of the sequence. Verifying that the sequence is geometric involves algebraically showing that the ratio between successive terms is constant.

In summary, algebra is the toolkit that allows us to manipulate and understand sequences and their behaviors on a deeper level.