The Geometric Series from the Finite Geometric Sequence

Oct 13, 2025 by magicalceylontours.com 128 views

Introduction to Geometric Series

A geometric series 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 series is:

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

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

Sigma Notation

Sigma notation is a compact way of writing a sum of terms. It is commonly used to represent a series or a sequence of numbers. The general form of sigma notation is:

$\sum_{i=1}^{n} a_i$

where 'a_i' is the ith term of the series, 'i' is the index of the term, and 'n' is the number of terms.

The Geometric Series from the Finite Geometric Sequence

The given finite geometric sequence is: 100, 80, 64.

To find the sigma notation for this geometric series, we need to find the common ratio 'r' and the first term 'a'.

Finding the Common Ratio

The common ratio 'r' can be found by dividing any term by its previous term. Let's divide the second term by the first term:

r = 80 / 100 = 0.8

Finding the First Term

The first term 'a' is given in the sequence: a = 100.

Writing the Sigma Notation

Now that we have the common ratio 'r' and the first term 'a', we can write the sigma notation for the geometric series.

The general form of a geometric series is:

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

Since we have a finite geometric sequence with 3 terms, we can write the sigma notation as:

$\sum_{n=1}^3 a r^{n-1}$

where 'a' is the first term, 'r' is the common ratio, and 'n' is the index of the term.

Option A: $\sum_{n=1}^3 100(3)^{n-1}$

This option is not correct because the common ratio is 0.8, not 3.

Option B: $\sum_{n=1}^3 100(0.8)^{n-1}$

This option is correct because the common ratio is 0.8, and the first term is 100.

Option C: $\sum_{n=1}^3 1.2(100)^{n-1}$

This option is not correct because the first term is 100, not 1.2.

Option D: $\sum_{n=1}^3 ...$

This option is not provided, so we cannot evaluate it.

Conclusion

The correct sigma notation for the geometric series from the given finite geometric sequence is:

$\sum_{n=1}^3 100(0.8)^{n-1}$

This option correctly represents the geometric series with a common ratio of 0.8 and a first term of 100.

Understanding the Importance of Sigma Notation

Sigma notation is a powerful tool for representing series and sequences. It allows us to write a sum of terms in a compact and concise way, making it easier to work with and analyze mathematical expressions.

Real-World Applications of Geometric Series

Geometric series have many real-world applications, including:

Finance: Geometric series are used to calculate compound interest and investment returns.
Physics: Geometric series are used to model the behavior of waves and oscillations.
Engineering: Geometric series are used to design and analyze electrical circuits and mechanical systems.

Common Misconceptions about Geometric Series

Some common misconceptions about geometric series include:

Geometric series are only for infinite sequences: Geometric series can also be used to represent finite sequences.
Geometric series are only for positive terms: Geometric series can also be used to represent sequences with negative terms.

Conclusion

In conclusion, the correct sigma notation for the geometric series from the given finite geometric sequence is:

$\sum_{n=1}^3 100(0.8)^{n-1}$

Q: What is a geometric series?

A: A geometric series 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 in the sequence.

Q: How do I find the common ratio?

A: To find the common ratio, divide any term by its previous term. For example, if the sequence is 2, 6, 18, the common ratio is 6/2 = 3.

Q: What is the first term?

A: The first term is the first number in the sequence.

Q: How do I write the sigma notation for a geometric series?

A: The sigma notation for a geometric series is:

$\sum_{n=1}^n a r^{n-1}$

where '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 series?

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

S_n = 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 nth term of a geometric series?

A: The formula for the nth term of a geometric series is:

a_n = a * r^(n-1)

where 'a_n' is the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.

Q: Can a geometric series have a negative common ratio?

A: Yes, a geometric series can have a negative common ratio. However, the terms of the series will alternate in sign.

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 sequence.

Q: Can a geometric series have a common ratio of 0?

A: No, a geometric series cannot have a common ratio of 0. This would result in a sequence of zeros.

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

A: A geometric series 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. An arithmetic series is a sequence of numbers where each term after the first is found by adding a fixed number called the common difference.

Q: How do I determine if a sequence is a geometric series?

A: To determine if a sequence is a geometric series, check if each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

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

A: To find the sum of a geometric series, use the formula:

S_n = 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: How do I find the nth term of a geometric series?

A: To find the nth term of a geometric series, use the formula:

a_n = a * r^(n-1)

where 'a_n' is the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.

Conclusion

In conclusion, geometric series are a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sigma notation for a geometric series is:

$\sum_{n=1}^n a r^{n-1}$

where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms. The formula for the sum of a geometric series is:

S_n = 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.