Find The Sum, If It Exists, Of The Infinite Geometric Series: $\[ 102 + 112.2 + 123.42 + \ldots \\]A. \[$ S = 1,020 \$\] B. \[$ S = 337.62 \$\] C. This Infinite Geometric Series Diverges. D. \[$ S = -1,020 \$\]

Understanding Infinite Geometric Series

An infinite geometric series is a type of mathematical series that consists of an infinite number of terms, where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. The series is said to be geometric if the ratio between consecutive terms is constant. In this article, we will explore how to find the sum of an infinite geometric series, and we will apply this concept to the given series: $102 + 112.2 + 123.42 + \ldots$

The Formula for the Sum of an Infinite Geometric Series

The sum of an infinite geometric series can be found using the formula: $S = \frac{a}{1 - r}$, where $a$ is the first term of the series and $r$ is the common ratio. This formula is valid only if the absolute value of the common ratio is less than 1, i.e., $|r| < 1$. If the absolute value of the common ratio is greater than or equal to 1, the series diverges, and the sum does not exist.

Applying the Formula to the Given Series

To find the sum of the given series, we need to identify the first term and the common ratio. The first term is $102$, and the common ratio can be found by dividing the second term by the first term: $r = \frac{112.2}{102} = 1.1$. Now, we can plug these values into the formula: $S = \frac{102}{1 - 1.1}$.

Calculating the Sum

To calculate the sum, we need to simplify the expression: $S = \frac{102}{1 - 1.1} = \frac{102}{-0.1} = -1020$. Therefore, the sum of the infinite geometric series is $-1020$.

Conclusion

In this article, we have explored the concept of infinite geometric series and how to find their sum using the formula: $S = \frac{a}{1 - r}$. We have applied this concept to the given series and found that the sum of the series is $-1020$. This result is consistent with option D.

Comparison with Other Options

Let's compare our result with the other options:

• Option A: $S = 1020$. This is incorrect, as the sum of the series is actually $-1020$.
• Option B: $S = 337.62$. This is also incorrect, as the sum of the series is $-1020$.
• Option C: This infinite geometric series diverges. This is incorrect, as the series converges and has a finite sum.

Conclusion

In conclusion, the sum of the infinite geometric series $102 + 112.2 + 123.42 + \ldots$ is $-1020$. This result is consistent with option D.

Final Answer

The final answer is: $\boxed{-1020}$

Understanding Infinite Geometric Series

An infinite geometric series is a type of mathematical series that consists of an infinite number of terms, where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. The series is said to be geometric if the ratio between consecutive terms is constant. In this article, we will explore some frequently asked questions about infinite geometric series.

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 of the series and $r$ is the common ratio. This formula is valid only if the absolute value of the common ratio is less than 1, i.e., $|r| < 1$.

Q: What happens if the absolute value of the common ratio is greater than or equal to 1?

A: If the absolute value of the common ratio is greater than or equal to 1, the series diverges, and the sum does not exist.

Q: How do I find the common ratio of an infinite geometric series?

A: To find the common ratio, you can divide any term by the previous term. For example, if the series is $102 + 112.2 + 123.42 + \ldots$, you can find the common ratio by dividing the second term by the first term: $r = \frac{112.2}{102} = 1.1$.

Q: What is the first term of an infinite geometric series?

A: The first term of an infinite geometric series is the first term of the series. For example, in the series $102 + 112.2 + 123.42 + \ldots$, the first term is $102$.

Q: Can I use the formula for the sum of an infinite geometric series if the series has a finite number of terms?

A: No, the formula for the sum of an infinite geometric series is only valid for infinite series. If the series has a finite number of terms, you can use the formula for the sum of a finite geometric series: $S = \frac{a(1 - r^n)}{1 - r}$, where $n$ is the number of terms.

Q: What is the difference between an infinite geometric series and a finite geometric series?

A: The main difference between an infinite geometric series and a finite geometric series is that an infinite geometric series has an infinite number of terms, while a finite geometric series has a finite number of terms.

Q: Can I use the formula for the sum of an infinite geometric series if the series has a common ratio of 1?

A: No, the formula for the sum of an infinite geometric series is only valid if the absolute value of the common ratio is less than 1. If the common ratio is 1, the series diverges, and the sum does not exist.

Conclusion

In this article, we have explored some frequently asked questions about infinite geometric series. We have discussed the formula for the sum of an infinite geometric series, how to find the common ratio, and what happens if the series diverges. We hope that this article has been helpful in understanding infinite geometric series.

Final Answer

The final answer is: $\boxed{S = \frac{a}{1 - r}}$