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Magazine Chapter 9: Problem 41
Use the formula for the sum of the first \(n\) terms of a geometric series to find the partial sum. $$ \sum_{n=1}^{10}-2 \cdot\left(\frac{1}{2}\right)^{n-1} $$
Short Answer
Expert verified
The partial sum is \(-\frac{1023}{256}\).
Step by step solution
01
Identify the Geometric Series Parameters
The geometric series is given by \(-2 \cdot \left(\frac{1}{2}\right)^{n-1}\). Here, the first term \(a = -2\), and the common ratio \(r = \frac{1}{2}\).
02
Recall the Sum Formula for a Geometric Series
The formula for the sum of the first \(n\) terms of a geometric series is \( S_n = a \frac{1-r^n}{1-r} \), where \(a\) is the first term and \(r\) is the common ratio.
03
Substitute the Values into the Formula
Substitute \(a = -2\), \(r = \frac{1}{2}\), and \(n = 10\) into the formula:\[ S_{10} = -2 \frac{1 - \left(\frac{1}{2}\right)^{10}}{1-\frac{1}{2}} \]
04
Simplify the Denominator
The denominator is \(1 - \frac{1}{2} = \frac{1}{2}\). So the expression becomes:\[ S_{10} = -2 \times \left( 2 \times \left( 1 - \left(\frac{1}{2}\right)^{10} \right) \right) \]
05
Calculate Powers and Perform Multiplications
Calculate \((\frac{1}{2})^{10} = \frac{1}{1024}\) and simplify further:\[ S_{10} = -2 \times 2 \times \left( 1 - \frac{1}{1024} \right) \]Which simplifies to:\[ S_{10} = -4 \times \left(\frac{1023}{1024}\right) \]
06
Derive the Unsimplified Result
Multiply to find the partial sum:\[ S_{10} = -4 \times \frac{1023}{1024} = -\frac{4092}{1024} \]
07
Simplify the Fraction
Simplify the fraction \(-\frac{4092}{1024}\) by dividing both numerator and denominator by 4:\[ S_{10} = -\frac{1023}{256} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Partial Sum
In a geometric series, the partial sum refers to the sum of a certain number of terms in the series. It's important because it allows us to calculate the total value of the sequence up to a specific point, rather than needing the entire infinite series. In the given sequence, we're interested in the sum of the first 10 terms. The partial sum, denoted as \(S_{10}\), is calculated using a specific formula that takes into account the first term, the common ratio, and the number of terms.
- The key is to break down the formula step by step.
- Substitute the values specific to your series correctly into the formula.
Common Ratio
The common ratio is a fundamental part of a geometric series. It's the factor by which each term is multiplied to get the next term in the series. In this example, the common ratio \(r\) is \(\frac{1}{2}\). Recognizing the common ratio is essential because it affects the growth or decay of the series as you go from one term to the next.
- It is calculated by dividing any term by its preceding term.
- In our series: The sequence starts with \(-2\) and continues with \(-1, -0.5, -0.25\), etc.
- Each term is half of the previous one, confirming our common ratio.
First Term
The first term of a geometric series is the starting point of the entire sequence. In our geometric series, the first term \(a\) is \(-2\). Knowing the first term is critical because it forms the base value from which all subsequent terms are generated using the common ratio.
- The first term gives the initial magnitude of the series.
- It is directly used in calculating the partial sum and is a key variable in the sum formula.
- The entire structure of the series builds upon this first term.
Sum Formula
The sum formula for a geometric series is an efficient way to calculate the sum of its terms up to a point without adding each term individually. The formula is:\[ S_n = a \frac{1-r^n}{1-r} \]where \(S_n\) is the partial sum, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
- Substitute known values into the formula to find the sum of the series.
- For our problem, \(a = -2\), \(r = \frac{1}{2}\), and \(n = 10\).
- The process involves calculating \(r^n\), finding the difference \(1-r^n\), and simplifying the expression using fraction arithmetic.
- A calculator might be helpful for managing fractional values and long expressions.
