Sum of a Geometric Sequence: A Step-by-Step Guide for SEO Optimization
Understanding the sum of a geometric sequence is crucial for many applications in mathematics, science, and engineering. In this article, we will explore how to find the sum of the first twelve terms of a specific geometric sequence, 3, -9, 27, -81, 243, and we will break down the process step-by-step.
Identifying the First Term and Common Ratio
The given geometric sequence starts with 3, -9, 27, -81, 243, and so on. The first term, denoted as a, is 3. To find the common ratio, r, we divide the second term by the first term:
- r -9 / 3 -3
Sum of the First 11 Terms
The formula for the sum of the first (n) terms of a geometric series is:
[S_n a cdot frac{1 - r^n}{1 - r} quad text{for } r eq 1 ]For the first 11 terms, we substitute (n 11), (a 3), and (r -3):
- S_{11} 3 cdot frac{1 - (-3)^{11}}{1 - (-3)}
Note that ((-3)^{11}) is a negative number because the exponent 11 is odd. Let's calculate the value:
- (-3)^{11} -177147
Now, substituting ((-3)^{11} -177147) into the formula:
- S_{11} 3 cdot frac{1 - (-177147)}{1 3} 3 cdot frac{1 177147}{4} 3 cdot frac{177148}{4} 3 cdot 44287 132861
Sum of the First 12 Terms
To find the sum of the first 12 terms, we use the same formula with (n 12):
- S_{12} 3 cdot frac{1 - (-3)^{12}}{1 - (-3)}
Since the exponent 12 is even, ((-3)^{12} 3^{12}), and we know that:
- 3^{12} 531441
Substituting (3^{12} 531441) into the formula:
- S_{12} 3 cdot frac{1 - 531441}{1 3} 3 cdot frac{-531440}{4} 3 cdot -132860 -398580
Conclusion
The sum of the first 12 terms of the geometric sequence 3, -9, 27, -81, 243, ... is (boxed{-398580}).
Understanding the process of finding the sum of a geometric sequence can help in various mathematical applications. Whether you're an SEO professional working on optimizing content, a student studying sequences and series, or a mathematician working on advanced calculations, this knowledge can be invaluable.