Problem 26 - 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: \(3,6,12,24, \ldots .\)

Step-by-Step Solution

Answer

The sum of the first 12 terms of the given geometric sequence is 12285.

Step 1: Identify the first term, the common ratio and the number of terms required

The first term of the sequence, \(a\) is given as 3. Each term of the sequence is twice the preceding term, hence, the common ratio, \(r\) of the sequence is 2. The number of terms required, \(n\) is given as 12.

Step 2: Substitute the values into the formula for the sum of a geometric sequence

Substitute \(a = 3\), \(r = 2\), and \(n = 12\) into the formula \(S_n = \frac{a(1 - r^n)}{1 - r}\).

Step 3: Calculate the sum of the sequence

Calculate to find that \(S_{12} = \frac{3(1 - 2^{12})}{1 - 2} = 12285\).

Key Concepts

Sum of a Geometric Sequence

A geometric sequence is a sequence 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. To find the sum of the first \(n\) terms, we use the formula:\[S_n = \frac{a(1 - r^n)}{1 - r}\]Here, \(S_n\) represents the sum of the first \(n\) terms. This formula is very handy for quickly finding the sum without needing to add each term one by one.

• Example Application: Suppose you have a sequence like 3, 6, 12, 24, and you want to know the sum of the first 12 terms. Using the formula can save a lot of time and effort.
• Important Note: This formula works only when the common ratio \(r eq 1\).

First Term

The first term in a geometric sequence is essential because it serves as the starting point for calculating subsequent terms. It's often denoted as \(a\).In the sequence given in the problem, the first term is 3. Recognizing this number as the first term helps in identifying other terms in the series.

• Role: The first term is directly used in the formula for the sum of the sequence.
• Tip: Always look for the initial number in the series to determine \(a\).

Common Ratio

The common ratio in a geometric sequence is the factor by which each term is multiplied to get the next term. It's usually represented as \(r\).In the example sequence 3, 6, 12, 24, each term is obtained by multiplying the previous term by 2. Thus, the common ratio \(r\) is 2.

• Identification: Divide any term by the previous term to determine the common ratio.
• Impact: The common ratio is crucial in shaping the progression and properties of the sequence.

n Terms

The number of terms \(n\) in a geometric sequence tells you how far the sequence extends. It's key to calculating the sum because it informs you how many terms you sum together.For the exercise at hand, \(n\) is given as 12, meaning you're summing up the first 12 terms of the sequence.

• Selection: This number depends on how many terms you want to cover or analyze.
• Application: Always ensure you substitute \(n\) correctly into the sum formula to avoid errors.