Question

Is the series geometric? If so, give the number of terms and the ratio between successive terms. If not, explain why not. $$ 5-10+20-40+80-\cdots-2560 $$

Short answer

If so, what is the common ratio and the number of terms? The series is geometric with a common ratio of -2 and 10 terms.

Step-by-step solution

Understand the given series

We are given the series: $$ 5 - 10 + 20 - 40 + 80 - \cdots - 2560 $$ Our job is to determine if this series is geometric or not.

Identify the ratio between successive terms

To determine if the series is geometric, we should find the ratio between successive terms. If this ratio is constant, then we can conclude that the series is geometric. To find the ratio between terms, we need to divide each term by its preceding term. Ratio between term 2 and term 1: $$ \frac{-10}{5} = -2 $$ Ratio between term 3 and term 2: $$ \frac{20}{-10} = -2 $$ Ratio between term 4 and term 3: $$ \frac{-40}{20} = -2 $$ Ratio between term 5 and term 4: $$ \frac{80}{-40} = -2 $$ The ratio between successive terms is a constant -2, so the series is geometric.

Calculate the number of terms in the series

The series goes from 5 to -2560, and we know that the common ratio is -2. To determine the number of terms, we will use the formula for the nth term of a geometric series: $$ a_n = a_1 \cdot r^{n-1} $$ Where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. In this case, \(a_1 = 5\), \(r = -2\), and \(a_n = -2560\). We need to find the value of \(n\) that satisfies this equation. Here's the equation with values plugged in: $$ -2560 = 5 \cdot (-2)^{n-1} $$

Solve for the number of terms

To solve for \(n\), we need to isolate it in the equation. First, divide both sides by 5: $$ \frac{-2560}{5} = (-2)^{n-1} $$ $$ -512 = (-2)^{n-1} $$ Now, since we have a power of -2, we can take the base 2 logarithm of both sides: $$ \log_2{512} = \log_2{(-2)^{n-1}} $$ When we simplify, we get: $$ 9 = (n-1) $$ Adding 1 to both sides, we find the value of n: $$ n = 10 $$

Summarize the results

Based on our calculations, the series: $$ 5 - 10 + 20 - 40 + 80 - \cdots - 2560 $$ is a geometric series with a common ratio of -2 and 10 terms.

Key concepts

Common Ratio

In a geometric series, one of the foundational concepts is the **common ratio**. This is the factor by which we multiply each term in the series to get the next term. It's essential to understand that the common ratio remains constant throughout the series. If you can find a consistent common ratio between each pair of successive terms, you're likely dealing with a geometric series.

For example, in the series provided: - 5, -10, 20, -40, 80, ..., -2560
Each term after the first is obtained by multiplying the previous term by -2. You can see this clearly:

• From 5 to -10: \(-10/5 = -2\)
• From -10 to 20: \(20/-10 = -2\)
• From 20 to -40: \(-40/20 = -2\)

And so on. Here, the common ratio is -2, making it easy to map out the entire sequence based on this repetition.

When examining series in your own math explorations, always check for a consistent pattern. This distinguishing feature informs you that the series is geometric.

Number of Terms

Determining the **number of terms** in a geometric series is a critical step after identifying the common ratio. This helps to understand how long the series extends, which is often needed for calculating the sum or verifying other properties of the series.

In the exercise example, you find the number of terms by using the formula for the nth term of a geometric series: \[ a_n = a_1 \times r^{n-1} \]
Here, \(a_n\) is the last term, \(a_1\) is the first term, and \(r\) is the common ratio. You are given the series ending at \(-2560\) with the first term as\(5\) and the common ratio as \(-2\). Thus, putting these in the formula:

\[ -2560 = 5 \times (-2)^{n-1} \]
Solving for \(n\), you divide first by 5, then use logarithmic operations to get:
\[ -512 = (-2)^{n-1} \]
\[ \log_2{512} = n-1 \]
This results in simplifying to \(9 = n - 1\) and then \(n = 10\).

Hence, there are 10 terms in this geometric series. Understanding this aspect allows for the evaluation of other properties like the sum or convergence of the series.

Algebra

The study of geometric series is deeply intertwined with **algebra**, as it involves manipulating expressions and solving equations to find specific terms or characteristics in a series. Algebra provides the tools necessary to derive formulas, work with variables, and break down complex problems into manageable steps.

In the context of solving the series problem, algebra helped in several ways:

• Establish the nth term equation: \(a_n = a_1 \times r^{n-1}\) used to determine the last term or find the number of terms.
• Manipulate expressions, such as dividing both sides of the equation by the initial term and using logarithms to solve powers of numbers.

For example, you divide to isolate \((-2)^{n-1}\) and then apply logarithms: \[ \frac{-2560}{5} = (-2)^{n-1} \]
\[ -512 = (-2)^{n-1} \]
\[ \log_2{512} = n-1 \]
This use of logarithms further demonstrates how algebraic skills can simplify and demystify the steps involved in solving geometric series problems.

Understanding these algebraic principles allows students to appreciate the structure and solve a variety of mathematical challenges beyond geometric series.