Problem 117 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence. $$a_{1}=100, r=e^{x}, 9 \text { th term }$$

Step-by-Step Solution

Answer

The ninth term of the geometric sequence is \(a_9 = 100 \times e^{8x}\).

Step 1: Identify the Values

From the exercise, we identify that \(a_1 = 100\), \(r = e^{x}\), and \(n = 9\). These values will be substituted into the geometric sequence formula to find the 9th term.

Step 2: Substitute Values into the Formula

We substitute the identified values into the formula. Thus, \(a_n = a_1 \times r^{(n-1)}\) becomes \(a_9 = 100 \times (e^{x})^{(9-1)}\).

Step 3: Simplify the Equation

Simplify the equation to get the final result. Note that one of the properties of exponents is that when you raise a power to a power, you multiply the exponents. Thus, \(a_9 = 100 \times e^{8x}\).

Key Concepts

Geometric Progression

A geometric progression, also known as a geometric sequence, is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For instance, in the sequence 2, 4, 8, 16, each term is obtained by multiplying the previous term by 2.

The general form of a geometric sequence can be written as:
\[a, ar, ar^2, ar^3, \ldots, ar^{n-1}\]
where \(a\) represents the first term, \(r\) is the common ratio, and \(n\) is the term number. The \(n\)th term of this sequence is given by the formula:
\[a_n = a \times r^{n-1}\]

In the exercise provided, we are working with the simple premise of a geometric progression where the first term \(a_1\) is 100 and the common ratio is given by the exponential function \(r = e^{x}\). To find the 9th term \(a_9\), we apply the geometric sequence formula using these values. This concept is fundamental to understanding how sequences are structured and how to navigate through them mathematically.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to the power of another number. This operation is represented as \(b^n\), where \(b\) is the base and \(n\) is the exponent. The exponent tells us how many times to multiply the base by itself. For example, \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).

Exponentiation has several important properties that simplify calculations:

• \(b^m \times b^n = b^{m+n}\)
• \((b^m)^n = b^{mn}\)
• \(b^{-n} = \frac{1}{b^n}\) for \(b eq 0\)
• \((ab)^n = a^n b^n\)

In our exercise, the exponential part comes into play with the common ratio \(r = e^{x}\), where \(e\) is Euler’s number, an irrational and transcendental number approximately equal to 2.71828. The term \(e^{x}\) represents the exponential growth factor. When calculating the 9th term, we utilized one of the exponent rules, \((b^m)^n = b^{mn}\), which in our case translated to \((e^{x})^{8} = e^{8x}\), significantly simplifying the process of finding the 9th term of the sequence.

Sequence and Series

In mathematics, the concepts of sequence and series are closely related but have distinct meanings. A sequence is an ordered list of numbers that follow a specific pattern or rule, while a series is the sum of the elements of a sequence. For example, if we have a sequence 1, 3, 5, 7, a series formed by these numbers would be 1 + 3 + 5 + 7.

A geometric sequence, as seen in the exercise, is a type of sequence. Whereas an arithmetic sequence increases by a constant difference between terms, a geometric sequence grows by a constant ratio—a multiplication factor. Series can use the terms of sequences to form their sums, such as the geometric series which adds the terms of a geometric sequence.
When looking at our exercise, we found the 9th term, but if we were to sum all terms from the 1st to the 9th, we would have a geometric series, which would be a separate calculation using the formula:
\[S_n = a \left(\frac{1 - r^n}{1 - r}\right)\]

Understanding the difference between sequences and series, and the relationship they share, can help grasp a variety of concepts in mathematics, from simple addition patterns to complex calculations involving growth and decay in functions and real-world phenomena.