Find The Common Ratio Of A Geometric Progression (GP) Whose Sum Of Infinite Terms Is 8, And Its Second Term Is 2.Solution:Let { A $}$ Be The First Term Of The GP. Then,1. { Ar = 2 $}$ (The Second Term)2. The Sum Of Infinite Terms

Introduction

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In this article, we will explore how to find the common ratio of a GP given that the sum of its infinite terms is 8 and its second term is 2.

The Formula for the Sum of Infinite Terms of a GP

The sum of an infinite geometric series is given by the formula:

$$S = \frac{a}{1 - r}$$

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

Given Information

We are given that the sum of the infinite terms of the GP is 8, so we can write:

$$8 = \frac{a}{1 - r}$$

We are also given that the second term of the GP is 2. Since the second term is obtained by multiplying the first term by the common ratio, we can write:

$$ar = 2$$

Solving for the Common Ratio

We can start by solving the equation $ar = 2$ for $a$. We can do this by dividing both sides of the equation by $r$:

$$a = \frac{2}{r}$$

Now, we can substitute this expression for $a$ into the equation $8 = \frac{a}{1 - r}$:

$$8 = \frac{\frac{2}{r}}{1 - r}$$

To simplify this expression, we can multiply both the numerator and the denominator by $r$:

$$8 = \frac{2}{r(1 - r)}$$

Now, we can cross-multiply to get rid of the fraction:

$$8r(1 - r) = 2$$

Expanding the left-hand side of the equation, we get:

$$8r - 8r^2 = 2$$

Rearranging the terms, we get:

$$8r^2 - 8r + 2 = 0$$

This is a quadratic equation in $r$. We can solve it using the quadratic formula:

$$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 8$, $b = -8$, and $c = 2$. Plugging these values into the formula, we get:

$$r = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(8)(2)}}{2(8)}$$

Simplifying the expression, we get:

$$r = \frac{8 \pm \sqrt{64 - 64}}{16}$$

The expression under the square root is equal to zero, so we can simplify the expression further:

$$r = \frac{8 \pm 0}{16}$$

This gives us two possible values for $r$:

$$r = \frac{8}{16} = \frac{1}{2}$$

$$r = \frac{0}{16} = 0$$

However, we know that the common ratio cannot be zero, since the GP would not be a GP in that case. Therefore, we can conclude that the common ratio is:

$$r = \frac{1}{2}$$

Conclusion

In this article, we have shown how to find the common ratio of a geometric progression (GP) given that the sum of its infinite terms is 8 and its second term is 2. We have used the formula for the sum of an infinite geometric series and the equation for the second term to solve for the common ratio. The common ratio is found to be $\frac{1}{2}$.

Example

Suppose we have a GP with a first term of 4 and a common ratio of $\frac{1}{2}$. We can use the formula for the sum of an infinite geometric series to find the sum of the infinite terms:

$$S = \frac{a}{1 - r} = \frac{4}{1 - \frac{1}{2}} = \frac{4}{\frac{1}{2}} = 8$$

This shows that the sum of the infinite terms of the GP is indeed 8.

Discussion

The common ratio of a GP is an important concept in mathematics, and it has many applications in fields such as finance, economics, and engineering. In this article, we have shown how to find the common ratio of a GP given that the sum of its infinite terms is 8 and its second term is 2. We have used the formula for the sum of an infinite geometric series and the equation for the second term to solve for the common ratio. The common ratio is found to be $\frac{1}{2}$.

References

• [1] "Geometric Progression" by Math Open Reference. Retrieved from https://www.mathopenref.com/geomprogression.html
• [2] "Infinite Geometric Series" by Khan Academy. Retrieved from https://www.khanacademy.org/math/precalculus/precalc-series-series-and-sequences/infinite-geometric-series/v/infinite-geometric-series

Further Reading

• "Geometric Progression" by Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Geometric_progression
• "Infinite Geometric Series" by Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Infinite_geometric_series

Code

import sympy as sp
a = sp.symbols('a')
r = sp.symbols('r')

eq1 = sp.Eq(a * r, 2)
eq2 = sp.Eq(8, a / (1 - r))

solution = sp.solve((eq1, eq2), (a, r))

print(solution)

Q: What is a Geometric Progression (GP)?

A: A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Q: What is the common ratio in a GP?

A: The common ratio is a fixed, non-zero number that is used to find each term in a GP. It is denoted by the letter 'r' and is used to multiply the previous term to get the next term.

Q: How do I find the common ratio of a GP?

A: To find the common ratio of a GP, you can use the formula for the sum of an infinite geometric series:

$$S = \frac{a}{1 - r}$$

where $a$ is the first term and $r$ is the common ratio. You can also use the equation for the second term:

$$ar = 2$$

to solve for the common ratio.

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:

$$S = \frac{a}{1 - r}$$

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

Q: How do I use the formula for the sum of an infinite geometric series to find the common ratio?

A: To use the formula for the sum of an infinite geometric series to find the common ratio, you can start by plugging in the values for the first term and the sum of the infinite terms. Then, you can solve for the common ratio.

Q: What is the equation for the second term of a GP?

A: The equation for the second term of a GP is:

$$ar = 2$$

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

Q: How do I use the equation for the second term to find the common ratio?

A: To use the equation for the second term to find the common ratio, you can start by plugging in the value for the second term. Then, you can solve for the common ratio.

Q: What is the relationship between the first term and the common ratio in a GP?

A: The relationship between the first term and the common ratio in a GP is given by the equation:

$$ar = 2$$

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

Q: How do I find the first term of a GP?

A: To find the first term of a GP, you can use the formula for the sum of an infinite geometric series:

$$S = \frac{a}{1 - r}$$

where $a$ is the first term and $r$ is the common ratio. You can also use the equation for the second term:

$$ar = 2$$

to solve for the first term.

Q: What is the relationship between the sum of an infinite geometric series and the common ratio?

A: The relationship between the sum of an infinite geometric series and the common ratio is given by the formula:

$$S = \frac{a}{1 - r}$$

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

Q: How do I use the formula for the sum of an infinite geometric series to find the common ratio?

A: To use the formula for the sum of an infinite geometric series to find the common ratio, you can start by plugging in the values for the first term and the sum of the infinite terms. Then, you can solve for the common ratio.

Q: What is the significance of the common ratio in a GP?

A: The common ratio is an important concept in mathematics, and it has many applications in fields such as finance, economics, and engineering. It is used to find each term in a GP, and it is also used to calculate the sum of an infinite geometric series.

Q: How do I apply the concept of the common ratio in real-world problems?

A: The concept of the common ratio can be applied in many real-world problems, such as calculating the growth or decay of a population, the interest on an investment, or the depreciation of an asset.

Q: What are some common applications of the common ratio?

A: Some common applications of the common ratio include:

• Calculating the growth or decay of a population
• Calculating the interest on an investment
• Calculating the depreciation of an asset
• Calculating the sum of an infinite geometric series

Q: How do I use the common ratio to calculate the sum of an infinite geometric series?

A: To use the common ratio to calculate the sum of an infinite geometric series, you can start by plugging in the values for the first term and the common ratio. Then, you can use the formula:

$$S = \frac{a}{1 - r}$$

to calculate the sum of the infinite series.

Q: What are some common mistakes to avoid when working with the common ratio?

A: Some common mistakes to avoid when working with the common ratio include:

• Assuming that the common ratio is always positive
• Assuming that the common ratio is always greater than 1
• Failing to check for convergence or divergence of the series
• Failing to use the correct formula for the sum of an infinite geometric series

Q: How do I check for convergence or divergence of a series?

A: To check for convergence or divergence of a series, you can use the ratio test or the root test. The ratio test involves calculating the limit of the ratio of consecutive terms, while the root test involves calculating the limit of the nth root of the nth term.

Q: What is the ratio test?

A: The ratio test is a method for checking for convergence or divergence of a series. It involves calculating the limit of the ratio of consecutive terms.

Q: What is the root test?

A: The root test is a method for checking for convergence or divergence of a series. It involves calculating the limit of the nth root of the nth term.

Q: How do I use the ratio test or the root test to check for convergence or divergence of a series?

A: To use the ratio test or the root test to check for convergence or divergence of a series, you can start by calculating the limit of the ratio of consecutive terms or the limit of the nth root of the nth term. If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive.

Q: What are some common applications of the ratio test and the root test?

A: Some common applications of the ratio test and the root test include:

• Checking for convergence or divergence of a series
• Calculating the sum of an infinite geometric series
• Calculating the sum of an infinite arithmetic series
• Calculating the sum of an infinite power series

Q: How do I use the ratio test and the root test in real-world problems?

A: The ratio test and the root test can be used in many real-world problems, such as calculating the growth or decay of a population, the interest on an investment, or the depreciation of an asset.

Q: What are some common mistakes to avoid when using the ratio test and the root test?

A: Some common mistakes to avoid when using the ratio test and the root test include:

• Failing to check for convergence or divergence of the series
• Failing to use the correct formula for the sum of an infinite geometric series
• Failing to use the correct formula for the sum of an infinite arithmetic series
• Failing to use the correct formula for the sum of an infinite power series

Q: How do I apply the concept of the ratio test and the root test in real-world problems?

A: The concept of the ratio test and the root test can be applied in many real-world problems, such as calculating the growth or decay of a population, the interest on an investment, or the depreciation of an asset.

Q: What are some common applications of the ratio test and the root test in finance?

A: Some common applications of the ratio test and the root test in finance include:

• Calculating the interest on an investment
• Calculating the depreciation of an asset
• Calculating the growth or decay of a population
• Calculating the sum of an infinite geometric series

Q: How do I use the ratio test and the root test in finance?

A: The ratio test and the root test can be used in finance to calculate the interest on an investment, the depreciation of an asset, the growth or decay of a population, and the sum of an infinite geometric series.

Q: What are some common mistakes to avoid when using the ratio test and the root test in finance?

A: Some common mistakes to avoid when using the ratio test and the root test in finance include:

• Failing to check for convergence or divergence of the series
• Failing to use the correct formula for the sum of an infinite geometric series
• Failing to use the correct formula for the sum of an infinite arithmetic series
• Failing to use the correct formula for the sum of an infinite power series

Q: How do I apply the concept of the ratio test and the root test in finance?