The given series is a finite geometric series represented by the summation notation: 5Σ(n=1)12(1/2)^n-1. Let's break down its components:...
- 5Σ: This part indicates that we are summing a series over a range of values for 'n'. The number '5' represents the number of terms we will be adding.
- (n=1): This part specifies the starting value of 'n' as 1, meaning we start the summation from the first term.
- 12(1/2)^n-1: This is the general formula for the nth term of the series. Here's how it works:
- 12: This is the first term of the series (when n = 1).
- (1/2)^n-1: This is the common ratio raised to the power of (n-1). The common ratio is (1/2), which means each term is half the previous one.
Calculating the Series
To evaluate the series, we simply need to sum the first five terms using the formula: 12(1/2)^n-1
Term 1 (n=1): 12(1/2)^(1-1) = 12(1/2)^0 = 12
Term 2 (n=2): 12(1/2)^(2-1) = 12(1/2)^1 = 6
Term 3 (n=3): 12(1/2)^(3-1) = 12(1/2)^2 = 3
Term 4 (n=4): 12(1/2)^(4-1) = 12(1/2)^3 = 1.5
Term 5 (n=5): 12(1/2)^(5-1) = 12(1/2)^4 = 0.75
Now, adding these terms together, we get:
5Σ(n=1)12(1/2)^n-1 = 12 + 6 + 3 + 1.5 + 0.75 = 23.25
Geometric Series Formula
While we calculated the series directly, there's a formula to calculate the sum of a finite geometric series:
S = a(1-r^n) / (1-r)
Where:
- S: Sum of the geometric series
- a: First term of the series
- r: Common ratio
- n: Number of terms
In our case, we have:
Plugging these values into the formula:
S = 12 (1 - (1/2)^5) / (1 - 1/2) = 12 (1 - 1/32) / (1/2) = 23.25
Conclusion
Both direct calculation and the geometric series formula lead to the same result: the sum of the series 5Σ(n=1)12(1/2)^n-1 is **23.25**. This demonstrates the power of mathematical formulas in efficiently solving problems involving series.
Detailed Prompt for Article Writing
**Title:** Evaluating the Series 5Σ(n=1)12(1/2)^n-1
**Target Audience:** Students and individuals learning about geometric series and summation notation.
**Keywords:**
* Geometric series
* Summation notation
* Finite series
* Common ratio
* Series evaluation
**Structure:**
1. **Introduction:**
* Briefly define geometric series and summation notation.
* State the problem: evaluate the series 5Σ(n=1)12(1/2)^n-1.
2. **Understanding the Series:**
* Break down the components of the series notation.
* Explain the role of each element: 5, Σ, (n=1), and 12(1/2)^n-1.
* Define the first term, common ratio, and number of terms in this specific series.
3. **Calculating the Series Directly:**
* Explain the process of calculating each term using the formula 12(1/2)^n-1.
* Show the calculation for each of the five terms (n=1 to n=5).
* Sum the calculated terms to find the total value of the series.
4. **Using the Geometric Series Formula:**
* Present the formula for the sum of a finite geometric series: S = a(1-r^n) / (1-r).
* Explain the meaning of each variable (S, a, r, n).
* Substitute the values from the given series (a = 12, r = 1/2, n = 5).
* Calculate the sum of the series using the formula.
5. **Comparison and Conclusion:**
* Compare the results obtained from direct calculation and the formula.
* Emphasize the efficiency of the formula for evaluating geometric series.
* Conclude by stating the final value of the series: 23.25.
**Optimization for Search Engines:**
* **Keywords:** Include the listed keywords throughout the article naturally, both in the body text and headings.
* **Headings:** Use for the title and for sub-headings related to the keywords.
* **Structure:** Organize the content logically with clear headings and sub-headings.
* **Readability:** Write in a clear, concise, and engaging style.
* **Formatting:** Use bullet points, numbered lists, and tags for code and formulas (with class "pre-scrollable").
**Additional Notes:**
* Use examples to illustrate the concepts.
* Avoid technical jargon and use plain language.
* Ensure the language is appropriate for the target audience.
* Proofread the article carefully for any errors in grammar or spelling.