Problem 29 - College Algebra

Question

Use the formula for the sum of the first n terms of a geometric sequence. Find the sum of the first 14 terms of the geometric sequence: \(-\frac{3}{2}, 3,-6,12, \ldots .\)

Step-by-Step Solution

Answer

The sum of the first 14 terms is \( -8191\).

Step 1: Identify the First Term, Common Ratio, and Number of Terms

From the sequence, we identify that the first term \(a = -\frac{3}{2}\), the common ratio \(r = -2\), and the number of terms \(n = 14\).

Step 2: Substitute into the Formula

Substitute \(a\), \(r\), and \(n\) into the formula \(S = a \frac{1-r^n}{1-r}\). This gives us \(S = -\frac{3}{2} \frac{1-(-2)^{14}}{1-(-2)}\).

Step 3: Evaluate the Expression

Evaluating the expression will give us our answer: evaluate \(-(-2)^{14}\), evaluate \(\frac{1}{1-(-2)}\), and then using this, evaluate \(S = -\frac{3}{2} \frac{1-(-2)^{14}}{1-(-2)}\). Do the calculations step by step and simplify to get the final sum.

Step 4: Perform the Calculation

Evaluating \(S = -\frac{3}{2} \frac{1-16384}{3}\) gives us the sum.

Key Concepts

Geometric Sequence Formula

Understanding the geometric sequence formula is like having a master key for unlocking patterns in mathematical sequences. A geometric sequence, also known as a geometric progression, 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.

The formula to find the nth term of a geometric sequence is given by: \[ a_n = a_1 \times r^{(n-1)} \] where:

• \( a_n \) is the nth term of the sequence,
• \( a_1 \) is the first term, and
• \( r \) is the common ratio.

For example, if you are given the sequence \( -\frac{3}{2}, 3, -6, 12, \ldots \), the first term \( a_1 = -\frac{3}{2} \) and the common ratio \( r = -2 \). By applying the formula, you can find any term in the sequence. This formula is essential for understanding the behavior of geometric sequences and is critical for solving problems that involve such sequences.

Sum of Geometric Series

Tackling the sum of a geometric series can feel a bit like a treasure hunt—once you know how to navigate the formula, you can uncover the sum of all terms with ease. The sum of the first n terms of a geometric series is calculated using the following formula: \[ S_n = a_1 \frac{1 - r^n}{1 - r} \] where:

• \( S_n \) is the sum of the first n terms,
• \( a_1 \) is the first term,
• \( r \) is the common ratio, and
• \( n \) is the number of terms to be added.

If we relate this to our earlier sequence example, to find the sum of the first 14 terms, we substitute \( a_1 = -\frac{3}{2} \), \( r = -2 \), and \( n = 14 \) into the formula to get \[ S_{14} = -\frac{3}{2} \frac{1 - (-2)^{14}}{1 - (-2)} \] Follow through with the calculations, and you'll discover the total sum of the series up to the 14th term. Remember that this formula only works when \( r \) does not equal 1, since that would result in a division by zero—a definite no-no in mathematics.

Common Ratio

The concept of a common ratio is akin to the heartbeat of a geometric sequence; it's the consistent rate at which each term is increased or decreased to form the next one. It's both simple and powerful—a single, unchanging number that defines the essence of the entire sequence.

Identifying the common ratio is quite straightforward. For any consecutive terms in the sequence, say \( a \) and \( b \), the common ratio \( r \) is found using the division: \[ r = \frac{b}{a} \] In our example sequence \( -\frac{3}{2}, 3, -6, 12, \ldots \), the common ratio is calculated by dividing any term by the one preceding it. Here, \( r = \frac{3}{-\frac{3}{2}} = -2 \). To find the common ratio is to understand the growth or decay pattern of the sequence. Whether the terms in your geometric sequence are soaring to the skies or diving into the depths, the common ratio is the numerical expression of this repetitive change.