Problem 19 - Precalculus with Limits

Question

In Exercises 17 - 28, write the first five terms of the geometric sequence \( a_1 = 1, r = \dfrac{1}{2} \)

Step-by-Step Solution

Answer

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

Step 1: Understanding Geometric Sequence

A geometric sequence is a sequence 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. The generic form of nth term of a geometric sequence is given as \( a_n = a_1 \cdot r^{(n-1)} \) where: \n- \( a_n \) is the nth term of the sequence, \n- \( a_1 \) is the first term, \n- \( r \) is the common ratio, and \n- \( n \) is the position of the term.

Step 2: Substitute the given values into the formula

Since we're given the first term \(a_1 = 1\) and the common ratio \(r = \frac{1}{2}\), we can substitute these values into the formula.

Step 3: Compute the first five terms

We're looking for the first five terms (i.e., when \(n=1, 2, 3, 4, 5\)). For each value of \(n\), substitute \(n\) into the equation and calculate the corresponding term: \n\nWhen \( n = 1 \), \(a_1 = 1 \cdot \left(\frac{1}{2}\right)^{1-1} = 1\), \nWhen \( n = 2 \), \(a_2 = 1 \cdot \left(\frac{1}{2}\right)^{2-1} = \frac{1}{2}\), \nWhen \( n = 3 \), \(a_3 = 1 \cdot \left(\frac{1}{2}\right)^{3-1} = \frac{1}{4}\), \nWhen \( n = 4 \), \(a_4 = 1 \cdot \left(\frac{1}{2}\right)^{4-1} = \frac{1}{8}\), \nWhen \( n = 5 \), \(a_5 = 1 \cdot \left(\frac{1}{2}\right)^{5-1} = \frac{1}{16}\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is the engine that drives the pattern of numbers. It is the consistent multiplier that connects each term to the next. When dealing with a geometric sequence, identifying this ratio is key to understanding how the sequence develops over time.

Let's consider our example, where the common ratio is given as \( r = \frac{1}{2} \). This means that to move from one term to the next, we multiply the current term by \( \frac{1}{2} \). As a result, the numbers get successively smaller because we are consistently halving the previous term. The power of the common ratio is evident as it shapes the entire sequence, whether it causes the sequence to grow, shrink, or even alternate based on its value being greater than 1, less than 1, or negative, respectively.

Sequence of Numbers

A sequence of numbers is essentially a list of numbers in a specific order, dominated by a rule that relates each number to its predecessors. In our context, the geometric sequence has a start, typically known as the first term, followed by a succession of terms each derived from the previous one by multiplying it by the common ratio.

If we were to visualize our example sequence with \( a_1 = 1 \) and \( r = \frac{1}{2} \) on a number line, we would see the terms \( 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} \) and so on, each spaced at intervals that shrink by half each time, which perfectly follows our predefined multiplication rule. This structured progression of numbers embodies a clear pattern that can be predicted, extended, or analyzed using the concepts of geometric sequences.

nth Term

The concept of the nth term is a way to refer to the term that appears in the nth position of a sequence without having to list out all preceding terms. For a geometric sequence, the formula to determine the nth term is \( a_n = a_1 \cdot r^{(n-1)} \), which encapsulates the effect of the common ratio raised to the power of one less than the term's position number.

For instance, in our given exercise, to find the third term (\(a_3\)), you would take the first term (\(a_1 = 1\)) and multiply it by the common ratio (\(r = \frac{1}{2}\)) to the power of two (\(3-1\)), resulting in \( \frac{1}{4} \). This nth term formula allows us to jump directly to any position in the sequence without laboriously multiplying each preceding term by the common ratio. It serves as a shortcut for both simple sequences like the one in our exercise, as well as complex sequences that you may encounter in higher-level mathematics.