Problem 54 - Algebra and Trigonometry

Question

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. $$ a_{n}=\left(\frac{1}{2}\right)^{n} $$

Step-by-Step Solution

Answer

The sequence \( a_{n} = \left(\frac{1}{2}\right)^{n} \) is a geometric sequence with a common ratio of \( \frac{1}{2} \).

Step 1: Understand the definition of Arithmetic and Geometric Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In a geometric sequence, the ratio between any two consecutive terms is constant. From the given sequence \( a_{n} = \left(\frac{1}{2}\right)^{n} \), it seems likely to be geometric.

Step 2: Check if the sequence is Geometric

In a geometric sequence, the ratio \( r \) between any two consecutive terms is constant. We can find this ratio by dividing a term by its preceding term. For \( a_{n} \), to check for \( a_{2} \) and \( a_{1} \), we would have \( \frac{a_{2}}{a_{1}} = \left(\frac{1}{2}\right)^{2} / \left(\frac{1}{2}\right)^{1} = \frac{1}{2} \). Checking for \( a_{3} \) and \( a_{2} \), \( \frac{a_{3}}{a_{2}} = \left(\frac{1}{2}\right)^{3} / \left(\frac{1}{2}\right)^{2} = \frac{1}{2} \). We see that the ratio is indeed constant, proving that \( a_{n} = \left(\frac{1}{2}\right)^{n} \) is a geometric sequence.

Step 3: Find the common ratio

From the previous step, the common ratio of the geometric sequence is \( r = \frac{1}{2} \). This ratio is the result of the division of any term by its preceding term.

Key Concepts

Common Ratio

In the exploration of sequences, the concept of a "common ratio" is central to understanding geometric sequences. A sequence is deemed geometric if the ratio between successive terms remains constant.
This constant is referred to as the common ratio.The sequence given, \( a_{n} = \left(\frac{1}{2}\right)^{n} \), is analyzed to determine its nature. A quick check involves dividing one term by its preceding term, such as \( \frac{a_{2}}{a_{1}} = \left(\frac{1}{2}\right)^{2} / \left(\frac{1}{2}\right)^{1} \). This yields a result of \( \frac{1}{2} \).This verification process consistently delivers the same ratio for every pair of consecutive terms, confirming the sequence's geometric nature:

• The ratio between \( a_{3} \) and \( a_{2} \) also results in \( \frac{1}{2} \).
• Consequently, the constant ratio, \( \frac{1}{2} \), establishes \( a_{n} \) as a geometric sequence.

Recognizing the common ratio is crucial, as it allows for the prediction of any term in the sequence based on its position.

Sequence Analysis

When determining the characteristics of a sequence, as with the given \( a_{n} = \left(\frac{1}{2}\right)^{n} \), sequence analysis is an essential tool. This involves systematically checking for properties typical of known sequence types.
Sequence analysis specifically targets identifying whether a sequence is arithmetic, geometric, or neither.This analysis aids in concluding not only the sequence type but also its parameters. For geometric sequences:

• One looks for a constant ratio between successive terms, indicative of a geometric nature.
• The calculation aligns with the definition, where every term (except the first) equals the previous term multiplied by the common ratio.

In analyzing our sequence, each ratio is constant, indicative of a geometric sequence. Proper sequence analysis simplifies complexes of otherwise intimidating sequences into understandable patterns.

Arithmetic Sequence

An arithmetic sequence offers a different approach from geometric sequences. Here, the focus is on the "common difference," not a "common ratio."In an arithmetic sequence, each term differs from the previous one by a constant amount. This common difference contrasts with the multiplicative nature seen in geometric sequences.
For example, in the sequence \( a_{n} \), the decision that it is not arithmetic comes from analyzing its structure, where multiplication by \( \frac{1}{2} \) forms subsequent terms rather than adding or subtracting a consistent figure.Key points of an arithmetic sequence include:

• A constant addition or subtraction known as the common difference.
• This property easily checked by subtracting consecutive terms, unlike a geometric series, where division is used.

Recognizing the distinction between arithmetic and geometric sequences is essential for correctly classifying sequences like \( a_{n} \) and understanding their behavior.