Problem 3 - Algebra and Trigonometry

Question

Write the first five terms of each geometric sequence. $$ a_{1}=20, \quad r=\frac{1}{2} $$

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are: 20, 10, 5, 2.5, 1.25.

Step 1: Identify the First Term and Common Ratio

The first term \( a_{1} \) is 20 and the common ratio \( r \) is \( \frac{1}{2} \) or 0.5.

Step 2: Apply the Formula

To get each subsequent term of the sequence, we multiply the previous term by the common ratio. Thus, the second term \( a_{2} \) will be \( a_{1} * r = 20 * \frac{1}{2} = 10 \).

Step 3: Find the Remaining Terms

Following the same logic, the remaining terms will be: \( a_{3} = a_{2} * r = 10 * \frac{1}{2} = 5 \), \( a_{4} = a_{3} * r = 5 * \frac{1}{2} = 2.5 \), \( a_{5} = a_{4} * r = 2.5 * \frac{1}{2} = 1.25 \).

Key Concepts

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply one term to get the next term. It is denoted by the letter 'r'. When the common ratio is positive, the sequence may either increase or decrease depending on whether the ratio is greater than or less than 1. If the common ratio is negative, this causes the sequence to alternate between positive and negative values. For example, in the provided exercise, the common ratio is \( \frac{1}{2} \), meaning each term is half the preceding term, resulting in a decreasing sequence.

It's crucial to distinguish the common ratio from the difference in an arithmetic sequence, where we add a constant to get to the next term. Understanding the common ratio is especially important because it defines the nature of the geometric sequence and also plays a key role in determining formulas for specific terms and the sum of terms in geometric series.

Geometric Sequence Formula

The formula for finding any term in a geometric sequence is given by \( a_{n} = a_{1} \times r^{(n-1)} \), where \( a_{n} \) represents the \( n^{th} \) term, \( a_{1} \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. This formula enables us to calculate the value of any term without having to list all previous terms.

In the exercise provided, to calculate the second term \( a_{2} \) after the first term which is 20, we would use the formula by setting \( n \) to 2, leading to \( a_{2} = 20 \times (\frac{1}{2})^{2-1} = 10 \). Similarly, for the fifth term, we would set \( n \) to 5, and so on. This formula is extremely powerful for computations, especially for high-value terms where manual multiplication would be impractical.

Sequence and Series

Understanding the distinction between sequences and series is essential for working with mathematical patterns. A 'sequence' is an ordered list of numbers that are typically defined by some function, as we've seen with the geometric sequence. In contrast, a 'series' refers to the sum of the terms of a sequence.

A geometric series, for instance, would involve summing the terms of a geometric sequence. This could look like \( a_{1} + a_{2} + a_{3} + ... + a_{n} \). For geometric series, there is a specific formula for finding the sum of the first \( n \) terms if the common ratio \( r eq 1 \): \( S_{n} = a_{1} \times \frac{1 - r^{n}}{1 - r} \). This formula allows you to quickly find the sum of a series without manually adding all terms.

Students often need to remember that series involve the operation of addition, whereas sequences are concerned with the ordering of terms based on a rule. Recognizing these differences helps to build a solid understanding of various mathematical concepts related to sequences and series.