Problem 12 - Precalculus with Limits

Question

In Exercises 9-32, write the first five terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = \left(\dfrac{1}{2} \right)^n \)

Step-by-Step Solution

Answer

The first five terms of the given sequence are: \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{8}\), \(\frac{1}{16}\), and \(\frac{1}{32}\).

Step 1: Apply the formula for n=1

Substitute n = 1 into the given formula: \(a_1 = \left(\frac{1}{2} \right)^1 = \frac{1}{2}\)

Step 2: Apply the formula for n=2

Substitute n = 2 into the formula: \(a_2 = \left(\frac{1}{2} \right)^2 = \frac{1}{4}\)

Step 3: Apply the formula for n=3

Next, substitute n = 3 into the formula: \(a_3 = \left(\frac{1}{2} \right)^3 = \frac{1}{8}\)

Step 4: Apply the formula for n=4

Put n = 4 into the formula: \(a_4 = \left(\frac{1}{2} \right)^4 = \frac{1}{16}\)

Step 5: Apply the formula for n=5

For last term, substitute n = 5 into the formula: \(a_5 = \left(\frac{1}{2} \right)^5 = \frac{1}{32}\)

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a base is raised to the power of a variable exponent. The general form is \( f(x) = a^x \), where \( a \) is a positive constant, and \( x \) is the exponent. In exponential functions, the variable is in the exponent rather than in the base, which represents repeated multiplication.
Exponential functions exhibit rapid growth or decay. As the variable increases, the value of the function either grows swiftly (if the base \( a \) is greater than 1) or decreases towards zero (if \( a \) is between 0 and 1).
In the exercise, the function \( a_n = \left(\frac{1}{2}\right)^n \) is an exponential decay function. The constant base \( \frac{1}{2} \) is less than 1, indicating that as \( n \) increases, the sequence values become smaller. This function helps in understanding processes like radioactive decay, cooling of objects, and population decreases.

Sequence and Series

Sequences and series are fundamental concepts in mathematics used to describe ordered lists of numbers. A sequence is essentially a list of numbers arranged in a specific order, according to a given rule. In contrast, a series is the sum of the elements of a sequence.
In our exercise, the numbers \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32} \) form a geometric sequence. A geometric sequence is characterized by each term being a constant multiple of the previous term. The factor by which each term is multiplied to get the next term is known as the "common ratio."
For this sequence, the common ratio is \( \frac{1}{2} \), meaning each term is half of the preceding term. Recognizing a sequence's type and its properties, like common ratio, helps in predicting future terms and analyzing convergences or divergences.

Mathematical Formula

A mathematical formula is a concise way of expressing information with symbols and constants. Formulas are used to describe certain mathematical relations and rules, enabling efficient problem-solving.
For the sequence \( a_n = \left(\frac{1}{2}\right)^n \), the formula provides a direct method to calculate any term in the sequence by substituting the position \( n \). This specific formula is an example of a closed-form expression, offering direct computation without needing preceding terms.
Understanding how to apply formulas facilitates working through problems methodically and enhances problem-solving efficiency. This particular formula, which employs exponential and repeated division principles, helps illustrate how formulas model consistent patterns or phenomena in real-world applications.