Problem 81 - Precalculus: Mathematical for Calculus

Question

If \(a_{1}, a_{2}\) \(a_{3}, \ldots\) is an arithmetic sequence with common difference \(d\) show that the sequence $$10^{a_{1}}, 10^{a_{2}}, 10^{a_{3}}, \dots$$ 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: Understanding the Arithmetic Sequence

An arithmetic sequence has the form \( a_1, a_2, a_3, \ldots \), where each term is given by \( a_n = a_1 + (n-1)d \). Here, \(d\) is the common difference between consecutive terms.

Step 2: Finding General Terms

From the arithmetic sequence, the \(n\)-th term is \( a_n = a_1 + (n-1)d \). Therefore, the \(n\)-th term of the new sequence is \( 10^{a_n} = 10^{a_1 + (n-1)d} \).

Step 3: Recognizing a Geometric Sequence

A sequence is geometric if the ratio of consecutive terms is constant. Compute the ratio \( \frac{10^{a_{n+1}}}{10^{a_n}} \).

Step 4: Calculating the Common Ratio

Using properties of exponents, \( \frac{10^{a_{n+1}}}{10^{a_n}} = \frac{10^{a_1 + nd}}{10^{a_1 + (n-1)d}} = 10^{a_1 + nd - a_1 - (n-1)d} = 10^d \).

Step 5: Conclusion on Common Ratio

The ratio \( 10^d \) is constant for all \(n\), confirming the sequence is geometric with common ratio \( 10^d \).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which each term increases by a constant amount, known as the common difference. For example, in the sequence 2, 4, 6, 8, 10, each number is obtained by adding 2 to the previous one. Here's how an arithmetic sequence can be understood:

• The first term is denoted by \(a_1\).
• Each subsequent term is calculated by adding the common difference \(d\) to the preceding term.

Mathematically, the \(n\)-th term of an arithmetic sequence can be expressed as \(a_n = a_1 + (n-1)d\). This structure makes it easy to find any term once you know the first term and the common difference.

Geometric Sequence

A geometric sequence is quite different from an arithmetic one. Here, each term is derived by multiplying the previous term by a constant known as the common ratio. Consider the sequence 3, 9, 27, 81. Each term is obtained by multiplying the preceding term by 3. Characteristics of a geometric sequence include:

• The first term, \(a_1\).
• Each next term is found by multiplying the previous term by the common ratio \(r\).

For example, the \(n\)-th term of a geometric sequence is given by \(a_n = a_1 \cdot r^{(n-1)}\). Understanding this helps in identifying sequences that are geometric and predicting upcoming terms.

Common Difference

The common difference in an arithmetic sequence plays a crucial role. It is the fixed amount added to each term to get the next term. To determine the common difference in a simple sequence, subtract the previous term from the current one. For instance, in the sequence 5, 10, 15, the common difference \(d\) is 5, because each term increases by 5.

Recognizing the common difference helps in constructing the rule \(a_n = a_1 + (n-1)d\), which is instrumental in solving problems related to arithmetic sequences and predicting unknown terms efficiently.

Common Ratio

In a geometric sequence, the common ratio is a constant multiplier used to find the subsequent term from the preceding one. It is crucial in identifying a sequence as geometric. To find the common ratio \(r\), simply divide any term by its previous term.

• Example: In the sequence 2, 6, 18, 54, the common ratio \(r\) is 3, since each term is obtained by multiplying the previous term by 3.
• Working with exponential sequences, like \(10^{a_1}, 10^{a_2}, 10^{a_3}, \ldots\), a sequence can transform into a geometric one where \(10^d\) becomes the common ratio if the original values form an arithmetic sequence.

Applying this knowledge guides the determination of the nature of sequences and aids in solving complex mathematical patterns.