Problem 2 - College Algebra

Question

Write the first five terms of each geometric sequence. $$ a_{1}=4, \quad r=3 $$

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are 4, 12, 36, 108, 324.

Step 1: Identify the first term and the common ratio

The first term \(a_1\) of the sequence is given as 4 and the common ratio \(r\) is 3.

Step 2: Calculation of the second term

Using the formula for the nth term of a geometric sequence, \(a_n = a_1 \cdot r^{(n-1)}\), we can calculate the second term as \(a_2 = 4 \cdot 3^{(2-1)} = 4 \cdot 3 = 12\).

Step 3: Calculation of the third term

Substitute n=3 into the formula, to get the third term as \(a_3 = 4 \cdot 3^{(3-1)} = 4 \cdot 9 = 36\).

Step 4: Calculation of the fourth term

Substitute n=4 into the formula, to get the fourth term as \(a_4 = 4 \cdot 3^{(4-1)} = 4 \cdot 27 = 108\).

Step 5: Calculation of the fifth term

Substitute n=5 into the formula, to get the fifth term as \(a_5 = 4 \cdot 3^{(5-1)} = 4 \cdot 81 = 324\).

Key Concepts

Common Ratio

The common ratio in a geometric sequence is perhaps the most crucial element that defines its nature. A geometric sequence, for those just beginning to encounter this term, is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio.

In the given exercise, the common ratio is denoted by the letter 'r', which is a consistent factor that determines how much one term will increase or decrease to get to the next. For instance, with a common ratio of 3, as seen in our exercise, each term is tripled as you move from one term to the next in the sequence. It’s the multiplication counterpart to the common difference in an arithmetic sequence, which instead uses addition.

Sequence Terms

Understanding sequence terms is essential in manipulating and studying geometric sequences. Each number in the sequence is called a term. In the provided exercise, the sequence begins with an initial term, commonly represented as \(a_1\).

This first term and the common ratio determine the entire sequence. All subsequent terms can be found by applying the common ratio successively to the first term. Notationally, if we want to refer to the third term of a sequence, we would use \(a_3\), the fourth term would be \(a_4\), and so forth. Recognizing the pattern of a geometric sequence can often give us a shortcut to finding later terms without calculating all the intervening steps, which is a handy trick for larger sequences.

Geometric Progression

A geometric progression is simply another name for a geometric sequence. These terms are often used interchangeably. A sequence of numbers is considered a geometric progression if the quotient of any two successive members of the sequence is a constant called the common ratio, as we've discussed.

This concept is not merely an abstract mathematical idea but has practical applications in areas such as computing, finance, biology, physics, and much more. Particularly in finance, geometric progressions are used to model exponential growth or decay, such as in calculating compound interest. Understanding how geometric sequences work can help us predict future growth, decay, and trends across various fields.