Problem 25 - College Algebra

Question

Use the formula for the sum of the first n terms of a geometric sequence. Find the sum of the first 12 terms of the geometric sequence: \(2,6,18,54, \ldots .\)

Step-by-Step Solution

Answer

The sum of the first 12 terms of the given geometric sequence is \(2(1 - 3^{12}) / (1 - 3) = 79766442.\)

Step 1: Identify the First Term and the Common Ratio

The first term (\(a\)) of the sequence is 2. The common ratio (\(r\)) can be found by dividing any term in the sequence by the preceding term. So, \(r = 6 / 2 = 3.\)

Step 2: Apply the Sum Formula

The number of terms (\(n\)) is given as 12. Substitute \(a = 2\), \(r = 3\), and \(n = 12\) into the sum formula - \(S_n = a(1 - r^n) / (1 - r)\). Carry out the calculations.

Step 3: Simplify the Result

Perform the necessary operations to arrive at the sum of the first 12 terms.

Key Concepts

Sum of a Geometric Series

Understanding the sum of a geometric series is crucial for grasping the essence of a geometric sequence. The formula for finding the sum, \( S_n \), of the first \( n \) terms of a geometric series is \( S_n = \frac{a(1 - r^n)}{1 - r} \) where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

For the given exercise, the first term is 2 and the common ratio — which is a consistent multiplier between terms — is 3. To find the sum of the first 12 terms, you plug these values into the formula and compute the result. The ability to identify these components correctly and apply the sum formula is fundamental in solving geometric series problems.

Common Ratio

The common ratio in a geometric progression plays a pivotal role in defining the character of the sequence. This ratio is determined by dividing a term in the sequence by its previous term, signifying the factor by which each term multiplies to arrive at the next.

For instance, with the sequence \(2, 6, 18, 54, \ldots\), you can identify the common ratio by dividing the second term by the first term, producing \(3\). This ratio remains constant throughout the progression, signifying that every subsequent term is three times its predecessor. Recognizing the common ratio is a key step in various calculations involving geometric progressions, such as finding the nth term or the sum of a series.

Geometric Progression

A geometric progression, or geometric sequence, is a string 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.

For example, the sequence in our exercise, \(2, 6, 18, 54, \ldots\), shows that each term is generated by multiplying the prior term by the common ratio of 3. A geometric progression can be increasing or decreasing, based on the value of the common ratio. If the common ratio is between \(-1\) and \(1\), the sequence will also be convergent or tend to a limit; otherwise, it will diverge.

Sequence and Series

The concepts of sequence and series are foundational in understanding various types of numerical patterns. A sequence is an ordered list of numbers, and a geometric sequence is a type of sequence where each term after the first is found by multiplying the preceding term by a constant value known as the common ratio.

A series, on the other hand, is the sum of the terms of a sequence. Specifically, a geometric series is the summation of a geometric sequence's terms. In the context of our problem, by adding the first 12 terms of the given geometric sequence, we are finding the value of a geometric series. Familiarity with these definitions and their distinctions is essential for proficiency in handling problems related to sequences and series.