Problem 31 - Algebra for College Students

Question

Find the sum of the first ten terms of the geometric sequence \(-4,8,-16,32, \ldots\). 1364

Step-by-Step Solution

Answer

The sum of the first ten terms is 1364.

Step 1: Identify the first term and the common ratio

The first term of the sequence, denoted as \( a \), is the first number in the sequence: \(-4\). The common ratio, \( r \), is found by dividing the second term by the first term: \( r = \frac{8}{-4} = -2 \).

Step 2: Write the formula for the sum of a geometric sequence

The sum of the first \( n \) terms of a geometric sequence can be calculated using the formula: \[ S_n = a \frac{1 - r^n}{1-r} \] where \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Step 3: Substitute the known values into the formula

We want to find the sum of the first ten terms, so \( n = 10 \). Substituting the values into the formula, we get:\[ S_{10} = -4 \frac{1 - (-2)^{10}}{1 - (-2)} \]

Step 4: Calculate \((-2)^{10}\) and simplify the equation

Calculate \((-2)^{10}\):\[ (-2)^{10} = 1024 \] Substitute back into the equation:\[ S_{10} = -4 \frac{1 - 1024}{1 + 2} \] This simplifies to:\[ S_{10} = -4 \frac{-1023}{3} \]

Step 5: Simplify the expression to find the sum

Calculate the expression:\[ -4 \times \frac{-1023}{3} = -4 \times -341 = 1364 \] Therefore, the sum of the first ten terms is 1364.

Key Concepts

Sum of Series

In mathematics, the sum of a series can often be intriguing and sometimes challenging. Specifically, for a geometric sequence, we use a particular formula to find the sum of the first few terms. A geometric sequence like

• -4
• 8
• -16
• 32
• ...

can change quickly and dramatically due to the nature of its progression.To determine the sum of the first ten terms, we use the formula: \[ S_n = a \frac{1 - r^n}{1-r} \]where:

• \( S_n \) represents the sum of the first \( n \) terms.
• \( a \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the total number of terms.

Substituting the values into this formula helps us determine the cumulative sum efficiently. This understanding is crucial when working with financial models, population studies, or computing technologies where such sequences often appear.

Common Ratio

The common ratio is a defining feature of a geometric progression. It indicates the factor by which each term in the sequence is multiplied to produce the next term. For example, given the sequence

• -4
• 8
• -16
• 32

the common ratio \( r \) is calculated as follows:\[ r = \frac{8}{-4} = -2 \]This ratio is consistent throughout the entire sequence and is vital in using the geometric series formula for sums. Recognizing and calculating the common ratio correctly ensures accurate analysis and utilization of geometric sequences in various mathematical and practical contexts.In many real-world situations, such as exponential growth or decay processes, identifying the common ratio helps analyze how quickly changes occur over time.

Geometric Progression

A geometric progression, or 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. The sequence

• -4
• 8
• -16
• 32

illustrates how quickly numbers can grow or shrink under geometric rules.Understanding geometric progression is crucial because it allows us to predict the long-term behavior of systems that follow such patterns. The formula for the general term in a geometric progression is:\[ a_n = a \cdot r^{n-1} \]where \( a_n \) is the \( n \)-th term, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. Whether we're calculating compound interest, analyzing population growth, or modeling natural phenomena, geometric progressions are indispensable tools in mathematics.