Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Writing the Terms of a Geometric Sequence In Exercises \(17-24,\) write the first five terms of the geometric sequence. $$a_{1}=6, r=3$$

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are \(6, 18, 54, 162, 486\).

Step 1: Identify the First Term and the Ratio

From the question, we are given the first term \(a_1 = 6\) and the ratio \(r = 3\). These values are constants and they can be used to generate the rest of the terms of the geometric sequence.

Step 2: Calculation of the Second Term

The second term of a geometric sequence is calculated by multiplying the first term by the ratio. So, the second term \(a_2\) can be calculated as: \(a_2 = a_1 \cdot r = 6 \cdot 3 = 18\).

Step 3: Calculation of the Third Term

Similarly, the third term \(a_3\) is calculated by multiplying the second term by the ratio. Thus, \(a_3 = a_2 \cdot r = 18 \cdot 3 = 54\).

Step 4: Calculation of the Fourth Term

Again, the fourth term \(a_4\) is calculated by multiplying the third term by the ratio. Hence, \(a_4 = a_3 \cdot r = 54 \cdot 3 = 162\.

Step 5: Calculation of the Fifth Term

Finally, the fifth term \(a_5\) is calculated by multiplying the fourth term by the ratio. As a result, \(a_5 = a_4 \cdot r = 162 \cdot 3 = 486\).

Key Concepts

Geometric Progression

Understanding geometric progression is essential for students grappling with sequence and series. A geometric progression, also known as 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. In simple terms, it's like a chain reaction where you start with an initial amount, and then consistently multiply by the same factor to get the next amount.

The general formula for the nth term of a geometric progression is given by: \(a_n = a_1 \cdot r^{(n-1)}\) where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

For example, if you have the first term of 6 and a common ratio of 3 as per our exercise, you’d get the sequence by consistently multiplying by 3: 6 (first term), 18 (second term), 54 (third term), and so on. The strength of a geometric progression lies in its predictability and its application across various fields such as finance, computer science, and physics.

Sequence and Series

The terms sequence and series often appear together but have distinct meanings in mathematics. A sequence is an ordered list of numbers that are typically identified by a pattern or rule, such as the geometric progression we discussed earlier. In contrast, a series is the sum of the terms of a sequence. Essentially, if you add up the numbers you've listed in a sequence, you're creating a series.

For clarity, let's consider our geometric sequence: 6, 18, 54, etc. If we add these terms together (6 + 18 + 54...), the result is a geometric series. Series can be finite or infinite, depending on whether the sequence has an end or continues indefinitely. Understanding series is important for real-life applications like calculating loan repayments or understanding the concept of infinity in theoretical mathematics.

Arithmetic Operations in Sequences

When working with sequences, arithmetic operations involve adding, subtracting, multiplying, or dividing each term by a constant value, or by another sequence’s corresponding terms. In the context of a geometric sequence, the primary operation is multiplication by the common ratio to get consecutive terms. This multiplication is the action that links each term in the sequence to its predecessor.

In more complex scenarios, one might perform other operations on sequences. For instance, multiplying two sequences might involve multiplying the nth term of one sequence by the nth term of another, resulting in a new sequence. Adding sequences similarly would mean adding the nth terms of both. Such operations manifest in various mathematical and practical contexts, underscoring the importance of mastering these fundamental concepts in algebra.