Problem 64 - Algebra and Trigonometry

Question

In Exercises \(63-64,\) find \(a_{2}\) and \(a_{3}\) for each geometric sequence. $$ 2, a_{2}, a_{3},-54 $$

Step-by-Step Solution

Answer

The second and third terms of the geometric sequence are \(a_2 = -6\) and \(a_3 = 18\).

Step 1: Identify the Common Ratio

Since this is a geometric sequence, there is a common ratio. Denote the common ratio as r. We are told the first term is 2 and the fourth term is -54. If we denote the terms of the sequence as \(a_1, a_2, a_3\), and \(a_4\), then we have that \(a_1 = 2\), \(a_4 = -54\), and, more generally, \(a_i = a_1 \times r^{i - 1}\). Substituting the values for \(a_4\) and \(a_1\), and simplifying gives us a cubic equation for the ratio \(r\): \(r^3 = \(-54\) / 2\) or \(r^3 = -27\).

Step 2: Solve for ratio r

Since \(r^3 = -27\), take the cube root of both sides to solve for r. In this case r equals to \(-3\).

Step 3: Find \(a_2\) and \(a_3\)

Use the formula for each term of a geometric sequence \(a_i = a_1 \times r^{i - 1}\). Substituting for \(a_2 = 2 \times (-3)^{2 - 1} = -6\) and \(a_3 = 2 \times (-3)^{3 - 1} = 18\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is the constant factor that each term is multiplied by to get the next term. It's what makes the sequence 'geometric.' Understanding the common ratio is critical for working with these sequences. It's like the 'DNA' of a geometric sequence, defining its unique characteristic and determining how it grows or shrinks.

Consider our example of a given sequence starting with 2 and ending with -54. To find the common ratio, we consider the relationship between consecutive terms. Since any term of the sequence can be obtained by multiplying the previous term by the common ratio, we can express this as an equation: \( a_{i} = a_{1} \times r^{i - 1} \). By applying this formula, the common ratio \( r \) can be calculated, which turns out to be the cube root of -27, or -3 in this case.

Sequence Terms

Individual elements in a geometric sequence are called sequence terms. They represent specific positions within the sequence and are denoted as \( a_{n} \) where \( n \) corresponds to their position. To find any term in a geometric sequence, you use the formula \( a_{n} = a_{1} \times r^{n - 1} \), where \( a_{1} \) is the first term and \( r \) is the common ratio. By plugging in the appropriate values for \( n \) and \( r \) and performing the necessary calculations, you can find any term in the sequence.

In our textbook example, once the common ratio is known, calculating \( a_{2} \) and \( a_{3} \) becomes straightforward: multiplying the first term, 2, by the common ratio raised to the power of one less than the term number. Hence, we find \( a_{2} = -6 \) and \( a_{3} = 18 \) by following this method.

Geometric Progression

A geometric progression is another name for a geometric sequence. It's a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This type of progression is distinguished by multiplying, rather than adding, a constant to get from one term to the next, unlike arithmetic progressions.

The progression has fascinating properties, making it applicable in various real-world scenarios, such as calculating compound interest, modeling exponential growth or decay, and even in some concepts of physics. The defining formula for any term in a geometric progression is \( a_{n} = a_{1} \times r^{n - 1} \), which ties together the initial term, the common ratio, and the position of any term in the progression.