Problem 89 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding Arithmetic Sequence

An arithmetic sequence is defined as a sequence of terms where the difference between successive terms is constant. This constant is known as the common difference, denoted by \(d\). Thus, the \(n\)-th term of an arithmetic sequence can be expressed as \(a_n = a_1 + (n-1)d\).

Step 2: Definition of Geometric Sequence

A geometric sequence is a sequence of terms where each term after the first term is found by multiplying the previous one by a fixed non-zero number called the common ratio \(r\). Mathematically, the sequence follows the form \(b_n = b_1 \times r^{n-1}\).

Step 3: Transforming Arithmetic to Geometric Sequence

Consider the sequence \(10^{a_1}, 10^{a_2}, 10^{a_3}, \ldots\). We seek to determine if this sequence is geometric. The terms translate to exponentiated form \(b_n = 10^{a_n}\). Since \(a_n = a_1 + (n-1)d\), we have \(b_n = 10^{a_1 + (n-1)d} = 10^{a_1} \times 10^{(n-1)d}\).

Step 4: Deriving the Common Ratio

Observe that \(b_n = 10^{a_1} \times 10^{(n-1)d}\). The expression \(10^{(n-1)d}\) determines the factor between terms of the sequence. The common ratio \(r\) can be obtained by simplifying \[\frac{b_{n+1}}{b_n} = \frac{10^{a_1+(n)d}}{10^{a_1+(n-1)d}} = \frac{10^{a_1 + nd}}{10^{a_1 + (n-1)d}} = 10^d.\] Hence, the common ratio of the sequence is \(10^d\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a type of number sequence where each term increases or decreases by a set, constant amount, known as the common difference. Each successive term in the sequence is reached by adding this fixed number to the previous term. This is a simple yet powerful concept in mathematics because it allows predicting any term of the sequence without knowing all preceding terms.

• The formula for the nth term in an arithmetic sequence is given as: \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.
• Arithmetic sequences are straightforward because the pattern of growth is linear and regular.

Imagine you're saving money, putting aside a fixed amount each month. The total savings form an arithmetic sequence, where each month's savings grow by the same amount.

Common Difference

In any arithmetic sequence, the magic number that links the terms together is called the common difference. It's the regular amount added or subtracted to get from one term to the next. Understanding the common difference is crucial because it defines the entire behavior of the sequence.

A positive common difference will make the sequence grow larger - think of climbing up stairs, each step takes you higher. Conversely, a negative common difference decreases the value of terms, akin to stepping down.

• If \( a_2 = a_1 + d \), the common difference \( d = a_2 - a_1 \).
• Finding \( d \) helps in determining any term in the sequence, thanks to the linear nature of arithmetic sequences.

For example, in the sequence 3, 5, 7, 9,... the common difference is 2. This consistent difference makes it easy to predict future values.

Common Ratio

The common ratio is a defining feature of a geometric sequence. It's the constant factor by which each term is multiplied to get to the next. This ratio is crucial because it controls the growth or decay pattern of the sequence.

While in an arithmetic sequence you add the common difference, in a geometric sequence, you multiply by the common ratio. This multiplicative pattern gives geometric sequences an exponential nature.

• The nth term in a geometric sequence can be expressed as \( b_n = b_1 \times r^{n-1} \), where \( b_1 \) is the first term and \( r \) is the common ratio.
• To find the common ratio \( r \), divide any term by the previous term: \( r = \frac{b_{n+1}}{b_n} \).

In the problem provided, by transforming an arithmetic base into an exponent (like \( 10^{a_n} \)), the sequence becomes geometric with the common ratio \( 10^d \), showing how surprisingly these sequences are interconnected.