Which Equation Could Be Used To Calculate The Sum Of The Geometric Series?$\frac{1}{3}+\frac{2}{9}+\frac{4}{27}+\frac{8}{81}+\frac{16}{243}$A.

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A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sum of a geometric series can be calculated using a specific equation, which is essential in various mathematical and real-world applications.

What is a Geometric Series?

A geometric series is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed number, known as the common ratio. The general form of a geometric series is:

a, ar, ar^2, ar^3, ...

where 'a' is the first term and 'r' is the common ratio.

Example of a Geometric Series

Let's consider the following geometric series:

13+29+427+881+16243\frac{1}{3}+\frac{2}{9}+\frac{4}{27}+\frac{8}{81}+\frac{16}{243}

In this series, the first term 'a' is 13\frac{1}{3}, and the common ratio 'r' is 23\frac{2}{3}.

Calculating the Sum of a Geometric Series

The sum of a geometric series can be calculated using the following equation:

S = a(1−rn)1−r\frac{a(1-r^n)}{1-r}

where 'S' is the sum of the series, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.

Derivation of the Equation

To derive the equation for the sum of a geometric series, we can use the following steps:

  1. Write the series as a sum of terms: a+ar+ar2+ar3+...+arn−1a + ar + ar^2 + ar^3 + ... + ar^{n-1}
  2. Multiply the series by 'r': ar+ar2+ar3+...+arn−1+arnar + ar^2 + ar^3 + ... + ar^{n-1} + ar^n
  3. Subtract the original series from the new series: (ar+ar2+ar3+...+arn−1+arn)−(a+ar+ar2+ar3+...+arn−1)(ar + ar^2 + ar^3 + ... + ar^{n-1} + ar^n) - (a + ar + ar^2 + ar^3 + ... + ar^{n-1})
  4. Simplify the expression: arn−aar^n - a
  5. Factor out 'a': a(rn−1)a(r^n - 1)
  6. Divide both sides by (1−r)(1-r): a(rn−1)1−r\frac{a(r^n - 1)}{1-r}

Simplifying the Equation

The equation for the sum of a geometric series can be simplified as follows:

S = a(1−rn)1−r\frac{a(1-r^n)}{1-r}

This equation can be used to calculate the sum of a geometric series, given the first term 'a', the common ratio 'r', and the number of terms 'n'.

Example: Calculating the Sum of a Geometric Series

Let's use the equation to calculate the sum of the geometric series:

13+29+427+881+16243\frac{1}{3}+\frac{2}{9}+\frac{4}{27}+\frac{8}{81}+\frac{16}{243}

In this series, the first term 'a' is 13\frac{1}{3}, the common ratio 'r' is 23\frac{2}{3}, and the number of terms 'n' is 5.

Plugging these values into the equation, we get:

S = 13(1−(23)5)1−23\frac{\frac{1}{3}(1-(\frac{2}{3})^5)}{1-\frac{2}{3}}

Simplifying the expression, we get:

S = 13(1−32243)13\frac{\frac{1}{3}(1-\frac{32}{243})}{\frac{1}{3}}

S = 1−322431-\frac{32}{243}

S = 211243\frac{211}{243}

Therefore, the sum of the geometric series is 211243\frac{211}{243}.

Conclusion

In conclusion, the sum of a geometric series can be calculated using the equation:

S = a(1−rn)1−r\frac{a(1-r^n)}{1-r}

This equation can be used to calculate the sum of a geometric series, given the first term 'a', the common ratio 'r', and the number of terms 'n'. The equation is derived by subtracting the original series from the new series, factoring out 'a', and dividing both sides by (1−r)(1-r).

Applications of Geometric Series

Geometric series have numerous applications in mathematics, finance, and real-world problems. Some of the applications include:

  • Finance: Geometric series are used to calculate the future value of an investment, the present value of a future amount, and the rate of return on an investment.
  • Mathematics: Geometric series are used to solve problems involving exponential growth and decay, and to calculate the sum of an infinite series.
  • Real-world problems: Geometric series are used to model population growth, the spread of diseases, and the decay of radioactive materials.

Common Ratio

The common ratio 'r' is a crucial component of a geometric series. It determines the rate of growth or decay of the series. If the common ratio is greater than 1, the series grows exponentially. If the common ratio is less than 1, the series decays exponentially.

Number of Terms

The number of terms 'n' in a geometric series determines the length of the series. It can be a finite number or an infinite number.

Infinite Geometric Series

An infinite geometric series is a geometric series with an infinite number of terms. The sum of an infinite geometric series can be calculated using the equation:

S = a1−r\frac{a}{1-r}

This equation can be used to calculate the sum of an infinite geometric series, given the first term 'a' and the common ratio 'r'.

Conclusion

In conclusion, geometric series are a fundamental concept in mathematics, and the sum of a geometric series can be calculated using the equation:

S = a(1−rn)1−r\frac{a(1-r^n)}{1-r}

Frequently Asked Questions

Q: What is a geometric series? A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: What is the common ratio? A: The common ratio is a fixed, non-zero number that is used to find each term in a geometric series. It determines the rate of growth or decay of the series.

Q: How do I calculate the sum of a geometric series? A: The sum of a geometric series can be calculated using the equation:

S = a(1−rn)1−r\frac{a(1-r^n)}{1-r}

where 'S' is the sum of the series, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.

Q: What is the formula for the sum of an infinite geometric series? A: The formula for the sum of an infinite geometric series is:

S = a1−r\frac{a}{1-r}

where 'S' is the sum of the series, 'a' is the first term, and 'r' is the common ratio.

Q: What is the difference between a finite geometric series and an infinite geometric series? A: A finite geometric series is a geometric series with a finite number of terms, while an infinite geometric series is a geometric series with an infinite number of terms.

Q: How do I determine the number of terms in a geometric series? A: The number of terms in a geometric series can be determined by counting the number of terms in the series, or by using the formula:

n = log⁡(Sa)log⁡(r)\frac{\log\left(\frac{S}{a}\right)}{\log(r)}

where 'n' is the number of terms, 'S' is the sum of the series, 'a' is the first term, and 'r' is the common ratio.

Q: What is the significance of the common ratio in a geometric series? A: The common ratio determines the rate of growth or decay of the series. If the common ratio is greater than 1, the series grows exponentially. If the common ratio is less than 1, the series decays exponentially.

Q: Can a geometric series have a common ratio of 1? A: Yes, a geometric series can have a common ratio of 1. In this case, the series is a constant series, and the sum of the series is equal to the first term.

Q: Can a geometric series have a common ratio of -1? A: Yes, a geometric series can have a common ratio of -1. In this case, the series is an alternating series, and the sum of the series depends on the number of terms.

Q: How do I determine if a geometric series converges or diverges? A: A geometric series converges if the absolute value of the common ratio is less than 1, and diverges if the absolute value of the common ratio is greater than or equal to 1.

Q: What is the relationship between the sum of a geometric series and the common ratio? A: The sum of a geometric series is equal to the first term divided by 1 minus the common ratio.

Q: Can a geometric series have a sum of 0? A: Yes, a geometric series can have a sum of 0. This occurs when the first term is 0, or when the common ratio is 1 and the number of terms is infinite.

Q: Can a geometric series have a sum of infinity? A: Yes, a geometric series can have a sum of infinity. This occurs when the common ratio is greater than 1 and the number of terms is infinite.

Conclusion

In conclusion, geometric series are a fundamental concept in mathematics, and the sum of a geometric series can be calculated using the equation:

S = a(1−rn)1−r\frac{a(1-r^n)}{1-r}

This equation can be used to calculate the sum of a geometric series, given the first term 'a', the common ratio 'r', and the number of terms 'n'. The equation is derived by subtracting the original series from the new series, factoring out 'a', and dividing both sides by (1−r)(1-r).