Problem 89 - Algebra and Trigonometry

Question

Exponentials of an Arithmetic Sequence If \(a_{1}, a_{2},\) \(a_{3}, \ldots\) is an arithmetic sequence with common difference \(d,\) show that the sequence $$ 10^{a_{1}}, 10^{a_{2}}, 10^{a_{3}}, \ldots $$ is a geometric sequence, and find the common ratio.

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of \(10^d\).

Step 1: Define the Arithmetic Sequence

An arithmetic sequence is given by \(a_1, a_2, a_3, \ldots\). Each term is defined by \(a_n = a_1 + (n-1) \cdot d\) where \(d\) is the common difference.

Step 2: Express the Exponential Sequence

The required sequence is \(10^{a_1}, 10^{a_2}, 10^{a_3}, \ldots\). For each term, \(10^{a_n} = 10^{a_1 + (n-1) \cdot d}\).

Step 3: Simplify the Exponential Term

By using the property of exponents that states \(a^{b+c} = a^b \cdot a^c\), express each term as \(10^{a_n} = 10^{a_1} \cdot 10^{(n-1) \cdot d}\).

Step 4: Verify the Geometric Nature

A sequence \(b_1, b_2, b_3, \ldots\) is geometric if the ratio of successive terms is constant. Compute the ratio \( \frac{10^{a_{n+1}}}{10^{a_n}} = \frac{10^{a_1 + n \cdot d}}{10^{a_1 + (n-1) \cdot d}} = 10^{d}\), which is a constant.

Step 5: Identify the Common Ratio

The common ratio \(r\) for the geometric sequence is \(10^d\) as shown in the previous step.

Key Concepts

Common Difference

In an arithmetic sequence, the term "common difference" refers to the consistent number you add to each term to arrive at the next term in the sequence. Let's say you have the sequence 2, 5, 8, 11, and so forth. Here, the common difference is 3 since you always add 3 to get from one term to the next. This is a crucial concept because it dictates the nature and behavior of arithmetic sequences.

• An arithmetic sequence's definition pivots around this constant: given the first term, you can generate any term with the formula \(a_n = a_1 + (n-1) \cdot d\).
• The ease of working with arithmetic sequences comes down to the constant addition of the common difference, making them linear in their general makeup.

Understanding the common difference is pivotal in solving sequences-related problems. It's the backbone that holds all the terms of an arithmetic sequence together, serving as the driving force behind the progression of the sequence.

Common Ratio

The common ratio is a key factor in geometric sequences. It is the constant factor by which you multiply each term to obtain the next term. For instance, in the geometric sequence 3, 9, 27, 81, the common ratio is 3. Each term is the previous term multiplied by 3.

• In this type of sequence, if you divide any term by its preceding term, the quotient is always the common ratio.
• The formula for a term in a geometric sequence is given by \(b_n = b_1 \cdot r^{(n-1)}\), where \(r\) is the common ratio.

In the exercise, when you transform an arithmetic sequence through exponentiation, the resulting sequence becomes geometric, with the common ratio being \(10^d\), where \(d\) is the common difference from the original arithmetic sequence. This conversion highlights how a consistent additive difference in arithmetic sequences translates into a multiplicative factor in geometric ones.

Exponential Functions

Exponential functions are a vital part of mathematical sequences, particularly when connecting arithmetic and geometric sequences. These functions deal with variables in the exponent, such as \(y = 10^x\).

• One of their unique properties is that exponential growth or decay is quite rapid compared to linear growth.
• They play an essential role in the transformation from arithmetic to geometric sequences, as seen in the exercise.

In our exercise, each term of the new sequence is formed by raising a fixed base (10, in this case) to the power of an arithmetic sequence term. This operation not only changes the nature of the sequence to geometric but is also scalable and predictable due to the properties of exponential functions. By understanding how to manipulate exponential functions, you gain the ability to view problems from multiple perspectives, finding ways to convert and interpret sequences accurately.