Problem 79 - 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 also geometric with a common ratio \( \frac{1}{r} \).

Step 1: Define the Original Geometric Sequence

A geometric sequence is one where each term is obtained by multiplying the previous term by a constant, known as the common ratio \( r \). Let the first term be \( a_1 = a \). Then, the sequence is: \( a, ar, ar^2, ar^3, \ldots \).

Step 2: Find Reciprocals of the Original Sequence

The sequence of reciprocals will be \( \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldots \) which equals \( \frac{1}{a}, \frac{1}{ar}, \frac{1}{ar^2}, \frac{1}{ar^3}, \ldots \).

Step 3: Examine the Reciprocals Sequence

Observe that the terms can be rewritten as: \( \frac{1}{a}, \frac{1}{a} \cdot \frac{1}{r}, \frac{1}{a} \cdot \frac{1}{r^2}, \frac{1}{a} \cdot \frac{1}{r^3}, \ldots \).

Step 4: Identify the New Common Ratio

From step 3, notice that each term is multiplied by \( \frac{1}{r} \) to get to the next. Thus, the sequence \( \frac{1}{a}, \frac{1}{ar}, \frac{1}{ar^2}, \ldots \) forms a geometric sequence with a common ratio of \( \frac{1}{r} \).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial element. It dictates how each subsequent term in the sequence is created. You can think of it as the "multiplier" that transforms one term into the next. For instance, if you have a sequence like 2, 4, 8, 16, and so on, the common ratio here is 2. This is because you multiply each term by 2 to get the next one.

When dealing with reciprocals of a geometric sequence, just like in our exercise, the concept of the common ratio remains essential. Following the example provided in the solution, always remember:

• The original sequence uses the common ratio \( r \).
• In the reciprocal sequence, the common ratio is \( \frac{1}{r} \).

This reciprocal transformation results in a new geometric sequence, helping maintain the energy and flow of recurring multiplication but in a reversed pattern.

Reciprocal Sequence

A reciprocal sequence is formed by taking the reciprocal of each term in the original sequence. It's like flipping each term over, just like a pancake!

In our initial example, if we have a geometric sequence like \( a, ar, ar^2, ar^3, \ldots \), its reciprocal sequence would take each term and turn it into \( \frac{1}{a}, \frac{1}{ar}, \frac{1}{ar^2}, \frac{1}{ar^3}, \ldots \). These new terms retain their own geometric structure, characterized by the reciprocal common ratio \( \frac{1}{r} \).

Here's the magical part: although we've flipped all the terms, the sequence formed by their reciprocals still holds the properties of a geometric sequence. This nifty transformation is useful for various mathematical analyses and can change the way we solve problems, often making them simpler and more approachable.

Mathematical Proof

When it comes to mathematics, proving concepts isn’t just about poking around until something sticks. It’s a logical process that builds up on established rules and previous findings. Take our problem, for example. We needed to prove that the sequence formed by reciprocals of a geometric sequence is itself geometric.

To do this, we identified the pattern in the reciprocal sequence: \( \frac{1}{a}, \frac{1}{ar}, \frac{1}{ar^2}, \ldots \). By precisely re-expressing it, each term can be seen to multiply by \( \frac{1}{r} \) to reach the next term, aligning perfectly with the definition of a geometric sequence.

Mathematical proofs require us to:

• Understand initial conditions or terms.
• Follow strict logical steps using known mathematical principles.
• Conclude by verifying that the sequence behaves as predicted by our proof.

This approach not only solidifies our understanding but also cements the integrity of mathematical reasoning, showing us exactly where, how, and why sequences fall into place.