Find The Sum Of The Geometric Series:$\sum_{i=1}^7 2(6)^{i-1}=$

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Introduction

In mathematics, 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 sum of a geometric series can be calculated using a formula, which is a powerful tool for solving problems involving geometric sequences. In this article, we will explore the concept of geometric series, understand the formula for calculating the sum, and apply it to find the sum of a specific geometric series.

What is a 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.

Example of a Geometric Series

Consider the following sequence:

2, 12, 72, 432, 2592, ...

This is a geometric series with a first term of 2 and a common ratio of 6. Each term is obtained by multiplying the previous term by 6.

Formula for Calculating the Sum of a Geometric Series

The sum of a geometric series can be calculated using 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.

Derivation of the Formula

To derive the formula for the sum of a geometric series, we can start with the definition of a geometric series:

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

We can write the sum of the first n terms as:

S_n = a + ar + ar^2 + ar^3 + ... + ar^(n-1)

We can factor out 'a' from each term:

S_n = a(1 + r + r^2 + r^3 + ... + r^(n-1))

We can recognize that the expression inside the parentheses is a geometric series with a first term of 1 and a common ratio of 'r'. We can use the formula for the sum of a geometric series to write:

S_n = a * (1 - r^n) / (1 - r)

Applying the Formula to Find the Sum of a Geometric Series

Now that we have the formula for the sum of a geometric series, we can apply it to find the sum of a specific geometric series. Let's consider the following geometric series:

∑i=172(6)i−1\sum_{i=1}^7 2(6)^{i-1}

This is a geometric series with a first term of 2 and a common ratio of 6. We can use the formula to find the sum of the first 7 terms:

S_7 = 2 * (1 - 6^7) / (1 - 6)

S_7 = 2 * (1 - 279936) / (-5)

S_7 = 2 * (-279935) / (-5)

S_7 = 559870

Therefore, the sum of the geometric series ∑i=172(6)i−1\sum_{i=1}^7 2(6)^{i-1} is 559870.

Conclusion

In this article, we explored the concept of geometric series, understood the formula for calculating the sum, and applied it to find the sum of a specific geometric series. Geometric series are a powerful tool for solving problems involving sequences, and the formula for calculating the sum is a valuable resource for mathematicians and scientists. We hope that this article has provided a clear understanding of geometric series and the formula for calculating the sum.

References

  • [1] "Geometric Series" by Math Open Reference
  • [2] "Geometric Series Formula" by Wolfram MathWorld
  • [3] "Geometric Series" by Khan Academy

Further Reading

  • [1] "Sequences and Series" by MIT OpenCourseWare
  • [2] "Calculus" by MIT OpenCourseWare
  • [3] "Mathematics for Computer Science" by MIT OpenCourseWare
    Geometric Series: Frequently Asked Questions =====================================================

Introduction

In our previous article, we explored the concept of geometric series, understood the formula for calculating the sum, and applied it to find the sum of a specific geometric series. In this article, we will answer some frequently asked questions about geometric series.

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, on the other hand, 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 series is geometric or arithmetic?

A: To determine if a series is geometric or arithmetic, you can look at the relationship between consecutive terms. If the ratio between consecutive terms is constant, then the series is geometric. If the difference between consecutive terms is constant, then the series is arithmetic.

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: How do I calculate the sum of a geometric series?

A: To calculate the sum of a geometric series, you can use the formula:

S_n = a * (1 - r^n) / (1 - r)

You can plug in the values of 'a', 'r', and 'n' into the formula to find the sum.

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

A: The common ratio in a geometric series determines the rate at which the terms increase or decrease. If the common ratio is greater than 1, the terms increase exponentially. If the common ratio is less than 1, the terms decrease 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 called an arithmetic series with a common difference of 0.

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

A: To find the sum of an infinite geometric series, you can use the formula:

S = 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 condition for the sum of an infinite geometric series to converge?

A: The sum of an infinite geometric series converges if and only if the absolute value of the common ratio is less than 1.

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 called an alternating geometric series.

Conclusion

In this article, we answered some frequently asked questions about geometric series. We hope that this article has provided a clear understanding of geometric series and the formula for calculating the sum.

References

  • [1] "Geometric Series" by Math Open Reference
  • [2] "Geometric Series Formula" by Wolfram MathWorld
  • [3] "Geometric Series" by Khan Academy

Further Reading

  • [1] "Sequences and Series" by MIT OpenCourseWare
  • [2] "Calculus" by MIT OpenCourseWare
  • [3] "Mathematics for Computer Science" by MIT OpenCourseWare