Problem 87 - Algebra and Trigonometry

Question

Reciprocals of a Geometric Sequence If \(a_{1}, a_{2}\) \(a_{3}, \ldots\) is a geometric sequence with common ratio \(r,\) show that the sequence $$ \frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}, \ldots $$ is also a geometric sequence, and find the common ratio.

Step-by-Step Solution

Answer

The sequence of reciprocals is a geometric sequence with common ratio \(\frac{1}{r}\).

Step 1: Define the Geometric Sequence

A geometric sequence is of the form \(a_1, a_2, a_3, \ldots\), where each term is multiplied by the common ratio \(r\) to get the next term. Therefore, the terms of the sequence are \(a_1, a_1r, a_1r^2, \ldots\).

Step 2: Express Reciprocals of the Sequence

The reciprocals of each term in the original sequence are \(\frac{1}{a_1}, \frac{1}{a_1r}, \frac{1}{a_1r^2}, \ldots\). These represent the new sequence we are analyzing.

Step 3: Identify the Common Numerator

Notice that in the sequence of reciprocals \(\frac{1}{a_1}, \frac{1}{a_1r}, \frac{1}{a_1r^2}, \ldots\), the numerator is always 1 and the denominators follow the original sequence structure \(a_1, a_1r, a_1r^2, \ldots\).

Step 4: Simplify the Consecutive Ratio

To show that the reciprocals form a geometric sequence, calculate the ratio of consecutive terms: \[\text{Ratio} = \frac{\frac{1}{a_1r}}{\frac{1}{a_1}} = \frac{\frac{1}{a_1r} \times a_1}{1} = \frac{1}{r}.\]

Step 5: Confirm a Geometric Structure

Since the ratio between consecutive terms in the reciprocal sequence is constant and equals \(\frac{1}{r}\), this sequence forms a geometric sequence. The common ratio of this new sequence is \(\frac{1}{r}\).

Key Concepts

Geometric Progression

A geometric progression, also referred to as a geometric sequence, is an ordered set of numbers where each term after the first is found by multiplying the preceding one by a fixed, non-zero number called the common ratio. For example, in the sequence 2, 4, 8, 16, ..., the common ratio is 2. This means each term is obtained by multiplying the previous term by 2.

• First Term: This is the starting point of the sequence, labeled as \(a_1\).
• Common Ratio: Denoted by \(r\), it determines the progression.
• General Term: The \(n\)-th term of a geometric sequence can be calculated via the formula \(a_n = a_1 \cdot r^{n-1}\).

The beauty of geometric sequences lies in their growth pattern, either exponential increase or decrease depending on whether the common ratio is greater than or less than 1.

Common Ratio

In any geometric progression, the common ratio \(r\) plays a crucial role. It's the factor by which you multiply a term to get the next one in the series. This ratio remains consistent throughout the sequence, which is why the pattern is predictable and uniform.

• If \(r > 1\), the sequence grows exponentially (e.g., 1, 3, 9, ... with \(r = 3\)).
• If \(0 < r < 1\), the sequence decreases (e.g., 100, 10, 1, ... with \(r = 0.1\)).
• If \(r = 1\), each term remains the same, resulting in a constant sequence.
• If \(r < 0\), the sequence alternates in sign, creating an oscillating pattern.

Understanding and identifying the common ratio is key to analyzing and solving problems related to geometric sequences. In our context, transitioning from the original sequence to its reciprocals,the common ratio becomes the fractional inverse \(\frac{1}{r}\). This reflects the consistent nature of the ratio across transformations.

Sequence of Reciprocals

A sequence of reciprocals is created by taking the reciprocal of each term in a given sequence. When applied to a geometric progression, the transformation of terms preserves the geometric nature but modifies the common ratio.
To illustrate, consider a geometric sequence \(a_1, a_1r, a_1r^2, \ldots\). The sequence of reciprocals formed by these terms will be \(\frac{1}{a_1}, \frac{1}{a_1r}, \frac{1}{a_1r^2}, \ldots\). Here, the terms of the new sequence are simply the inverses of the original sequence.
What's intriguing is that this sequence of reciprocals also forms a geometric progression. However, the new common ratio turns into \(\frac{1}{r}\), effectively reversing the order of exponential growth or decay seen in the original sequence. This is a fundamental concept in understanding how geometric relationships carry through reciprocal transformations while maintaining the sequence's intrinsic properties.