Find The Sum Of The First 50 Terms Of The Sequence: \[$-6, -2, 2, 6, \ldots\$\]

Introduction

In mathematics, a geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The given sequence ${-6, -2, 2, 6, \ldots}$ is a geometric sequence with a common ratio of ${-\frac{1}{3}}$. In this article, we will discuss how to find the sum of the first 50 terms of this sequence.

Understanding Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric sequence is:

$$a, ar, ar^2, ar^3, \ldots$$

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

Finding the Sum of a Geometric Sequence

The sum of the first $n$ terms of a geometric sequence can be found using the formula:

$$S_n = \frac{a(1-r^n)}{1-r}$$

where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Applying the Formula to the Given Sequence

In the given sequence ${-6, -2, 2, 6, \ldots}$, the first term $a$ is $-6$ and the common ratio $r$ is $-\frac{1}{3}$. We want to find the sum of the first 50 terms, so we will use the formula with $n=50$.

Calculating the Sum

Plugging in the values, we get:

$$S_{50} = \frac{-6\left(1-\left(-\frac{1}{3}\right)^{50}\right)}{1-\left(-\frac{1}{3}\right)}$$

Simplifying the Expression

To simplify the expression, we can start by evaluating the exponent:

$$\left(-\frac{1}{3}\right)^{50} = \frac{1}{3^{50}}$$

Continuing the Simplification

Now, we can substitute this value back into the expression:

$$S_{50} = \frac{-6\left(1-\frac{1}{3^{50}}\right)}{1+\frac{1}{3}}$$

Further Simplification

To simplify the expression further, we can multiply the numerator and denominator by $3$:

$$S_{50} = \frac{-18\left(1-\frac{1}{3^{50}}\right)}{3+1}$$

Final Simplification

Now, we can simplify the expression by combining the terms in the numerator:

$$S_{50} = \frac{-18\left(\frac{3^{50}-1}{3^{50}}\right)}{4}$$

Final Calculation

To calculate the final value, we can multiply the numerator and denominator by $3^{50}$:

$$S_{50} = \frac{-18(3^{50}-1)}{4 \cdot 3^{50}}$$

Final Answer

After simplifying the expression, we get:

$$S_{50} = \frac{-18(3^{50}-1)}{4 \cdot 3^{50}} = \frac{-18}{4} = -\frac{9}{2}$$

Conclusion

In this article, we discussed how to find the sum of the first 50 terms of a geometric sequence. We used the formula for the sum of a geometric sequence and applied it to the given sequence. We simplified the expression and calculated the final value to get the answer.

Real-World Applications

Geometric sequences have many real-world applications, such as:

• Finance: Geometric sequences can be used to model the growth of investments or the decay of debts.
• Biology: Geometric sequences can be used to model the growth of populations or the decay of radioactive materials.
• Computer Science: Geometric sequences can be used to model the growth of algorithms or the decay of data.

Future Research

There are many areas of future research in geometric sequences, such as:

• Developing new formulas: Developing new formulas for the sum of geometric sequences can help to simplify calculations and make them more efficient.
• Applying geometric sequences to new fields: Applying geometric sequences to new fields, such as physics or engineering, can help to model complex phenomena and make predictions.
• Investigating the properties of geometric sequences: Investigating the properties of geometric sequences, such as their convergence or divergence, can help to understand their behavior and make predictions.

References

• "Geometric Sequences" by Math Open Reference
• "Geometric Sequences and Series" by Khan Academy
• "Geometric Sequences" by Wolfram MathWorld

Glossary

• Geometric sequence: A sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
• Common ratio: The fixed, non-zero number that is multiplied by each term to get the next term.
• Sum of a geometric sequence: The sum of the first $n$ terms of a geometric sequence.
• Formula for the sum of a geometric sequence: The formula used to calculate the sum of the first $n$ terms of a geometric sequence.

Conclusion

In conclusion, geometric sequences are a powerful tool for modeling complex phenomena and making predictions. By understanding the properties of geometric sequences and applying them to real-world problems, we can gain a deeper understanding of the world around us.

Introduction

In our previous article, we discussed how to find the sum of the first 50 terms of a geometric sequence. In this article, we will answer some frequently asked questions about geometric sequences.

Q: What is a geometric sequence?

A: A geometric sequence is a sequence of numbers where 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 the fixed, non-zero number that is multiplied by each term to get the next term.

Q: How do I find the sum of a geometric sequence?

A: To find the sum of a geometric sequence, you can use the formula:

$$S_n = \frac{a(1-r^n)}{1-r}$$

where $S_n$ is the sum of the first $n$ terms, $a$ 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 a geometric sequence?

A: The formula for the sum of a geometric sequence is:

$$S_n = \frac{a(1-r^n)}{1-r}$$

Q: How do I apply the formula to a geometric sequence?

A: To apply the formula to a geometric sequence, you need to know the first term, the common ratio, and the number of terms. You can then plug these values into the formula to find the sum.

Q: What are some real-world applications of geometric sequences?

A: Geometric sequences have many real-world applications, such as:

• Finance: Geometric sequences can be used to model the growth of investments or the decay of debts.
• Biology: Geometric sequences can be used to model the growth of populations or the decay of radioactive materials.
• Computer Science: Geometric sequences can be used to model the growth of algorithms or the decay of data.

Q: What are some areas of future research in geometric sequences?

A: There are many areas of future research in geometric sequences, such as:

• Developing new formulas: Developing new formulas for the sum of geometric sequences can help to simplify calculations and make them more efficient.
• Applying geometric sequences to new fields: Applying geometric sequences to new fields, such as physics or engineering, can help to model complex phenomena and make predictions.
• Investigating the properties of geometric sequences: Investigating the properties of geometric sequences, such as their convergence or divergence, can help to understand their behavior and make predictions.

Q: What are some common mistakes to avoid when working with geometric sequences?

A: Some common mistakes to avoid when working with geometric sequences include:

• Not checking for convergence: Geometric sequences can converge or diverge, depending on the common ratio. Make sure to check for convergence before applying the formula.
• Not using the correct formula: Make sure to use the correct formula for the sum of a geometric sequence.
• Not plugging in the correct values: Make sure to plug in the correct values for the first term, common ratio, and number of terms.

Q: What are some resources for learning more about geometric sequences?

A: Some resources for learning more about geometric sequences include:

• Math Open Reference: A online reference book that covers geometric sequences and other mathematical topics.
• Khan Academy: A online learning platform that covers geometric sequences and other mathematical topics.
• Wolfram MathWorld: A online reference book that covers geometric sequences and other mathematical topics.

Conclusion

In conclusion, geometric sequences are a powerful tool for modeling complex phenomena and making predictions. By understanding the properties of geometric sequences and applying them to real-world problems, we can gain a deeper understanding of the world around us.