Problem 25 - College Algebra

Question

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 12 terms of the geometric sequence: $$2,6,18,54, \dots$$

Step-by-Step Solution

Answer

The sum of the first 12 terms of the given geometric sequence is \[S = \frac{2 \cdot (3^{12} - 1)}{(3 - 1)}.\] To get the numerical value, this would require a calculator.

Step 1: Identify the first term and common ratio

From the sequence, we can see that the first term \(a\) is 2 and the common ratio \(r\) is 3. The ratio is computed by dividing any term in the sequence by the preceding term.

Step 2: Insert the values into the Sum formula

Using the formula for the sum of a geometric sequence \(S = \frac{a(r^n - 1)}{r - 1}\), we can insert our values \(a = 2\), \(r = 3\), and \(n = 12\). So, \(S = \frac{2(3^{12} - 1)}{(3 - 1)}\).

Step 3: Calculate the Sum

Now we just need to do the calculation. This yields the sum of the first 12 terms of the sequence.

Key Concepts

Sum Formula

The sum formula for a geometric sequence allows us to find the total of a specific number of terms in the sequence without having to add each term manually. For a geometric sequence, the sum of the first \( n \) terms, denoted as \( S_n \), is given by the formula:

• \( S_n = \frac{a(r^n - 1)}{r - 1} \)

Here:

• \( a \) is the first term of the sequence.
• \( r \) is the common ratio, which is a fixed number that each term is multiplied by to get the next term.
• \( n \) is the number of terms for which we want to find the sum.

For the sequence 2, 6, 18, 54, and so on, the sum of the first 12 terms is calculated using this formula by substituting the known values for \( a \), \( r \), and \( n \). This effectively speeds up the process and ensures accuracy by working directly with the primary mathematical relationship governing the sequence.

First n Terms

When we refer to the 'first \( n \) terms' in a geometric sequence, we are specifying a particular portion of the sequence that includes the initial \( n \) terms starting from the first term. Knowing how to identify and handle this section of the sequence is crucial for applying the sum formula correctly.

• For the given sequence, we were asked to sum the first 12 terms. This means starting with the first term, 2, and including all subsequent terms up to the 12th term.
• In general, identifying the first \( n \) terms is straightforward: count from the first term to the \( n \)-th term in the sequence.
• Using the sum formula, we focus only on these \( n \) terms, regardless of the total number of terms present in the entire sequence.

This approach simplifies the calculation significantly when dealing with infinite or very long sequences, as it limits our scope of interest to just the parts that matter for the problem at hand.

Common Ratio

The common ratio in a geometric sequence is a crucial element as it dictates the progression of the terms. The common ratio, \( r \), is a constant factor by which each term is multiplied to obtain the next term in the sequence.

• For example, in the sequence 2, 6, 18, 54, we find the common ratio by dividing any term by its preceding term. Here, \( r = 6 \div 2 = 3 \).
• Since every term is the result of multiplying the previous term by this common ratio, consistency in the value of \( r \) is what classifies the sequence as geometric.
• This ratio must not be zero, as a common ratio of zero would cause all terms after the first to become zero.

Understanding the common ratio helps in applying the sum formula correctly, ensuring each term's contribution to the sequence's sum is appropriately calculated.