Worked examples: finite geometric series | High School Math | Khan Academy

Khan Academy
2 Aug 201606:54
EducationalLearning
32 Likes 10 Comments

TLDRThe video script explores the concept of summing finite geometric series, starting with a series where each term is multiplied by 10/11. It demonstrates the formula for the sum of the first 50 terms, emphasizing the importance of identifying the common ratio. The script then delves into another series with a common ratio of -0.99, clarifying the difference between the terms and the power to which the ratio is raised. Finally, it examines a recursively defined series with a common ratio of 9/10, illustrating how to calculate the sum of the first 30 terms. The script is an educational walkthrough of geometric series, highlighting the process of identifying terms and applying the sum formula.

Takeaways
  • 📚 The script discusses the process of finding the sum of the first 50 terms of a geometric series where each term is obtained by multiplying the previous term by 10/11.
  • 🔍 The formula for the sum of a finite geometric series is applied, which is the first term multiplied by one minus the common ratio to the power of the number of terms, divided by one minus the common ratio.
  • 🔢 The common ratio for the first series is 10/11, and the formula is simplified to 11 times one minus (10/11) raised to the 50th power, divided by 1.
  • 🤔 The second series discussed involves a change in sign, with a common ratio of -0.99, and the series is summed up to the 80th term, not the 79th, as the zeroth power term is included.
  • 📈 The sum for the second series is calculated using the formula with the common ratio raised to the 80th power and then divided by 1.99.
  • 📉 The third series is defined recursively, with each term being the previous term multiplied by 9/10, and the sum of the first 30 terms is calculated accordingly.
  • 📝 The importance of correctly identifying the common ratio and the number of terms in a series is emphasized to avoid calculation errors.
  • 🧩 The script provides a step-by-step breakdown of how to apply the geometric series sum formula to different scenarios, including handling negative common ratios.
  • 📊 The script illustrates the process of simplifying the formula and emphasizes the use of parentheses to ensure correct calculations.
  • 📘 The arithmetic involved in these calculations can become complex, and the script suggests that while simplification is helpful, a calculator is often needed for exact values.
  • 🔑 The script concludes by highlighting the importance of understanding the recursive nature of series and how it affects the calculation of their sums.
Q & A

  • What type of series is being discussed in the script?

    -The script discusses a geometric series, where each term is found by multiplying the previous term by a common ratio.

  • What is the common ratio for the first series mentioned in the script?

    -The common ratio for the first series is 10/11.

  • How is the sum of a finite geometric series calculated?

    -The sum of a finite geometric series is calculated using the formula: \( S_n = \frac{a_1(1 - r^n)}{1 - r} \), where \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

  • What is the first term of the series discussed in the script?

    -The first term of the series discussed in the script is 1.

  • How many terms are in the first series mentioned in the script?

    -The first series mentioned in the script has 50 terms.

  • What is the common ratio for the second series discussed in the script?

    -The common ratio for the second series is -0.99.

  • How many terms are in the second series mentioned in the script?

    -The second series mentioned in the script has 80 terms.

  • What is the significance of the negative sign in the common ratio of the second series?

    -The negative sign in the common ratio indicates that the terms alternate in sign, which affects the calculation of the sum of the series.

  • What is the first term of the second series discussed in the script?

    -The first term of the second series is not explicitly stated in the script, but it is implied to be 1 since it is 0.99 to the zeroth power.

  • What is the recursive definition of the series in the third example of the script?

    -In the third example, the series is defined recursively where each term \( A_n \) is equal to the previous term \( A_{n-1} \) multiplied by 9/10.

  • How is the sum of the first 30 terms of the recursive series calculated in the script?

    -The sum of the first 30 terms of the recursive series is calculated using the formula for the sum of a finite geometric series, with the common ratio being 9/10.

Outlines
00:00
🔢 Sum of a Geometric Series

The first paragraph introduces a problem of calculating the sum of the first 50 terms of a geometric series where each term is obtained by multiplying the previous term by 10/11. The formula for the sum of a finite geometric series is applied, which is the first term multiplied by one minus the common ratio raised to the power of the number of terms, all divided by one minus the common ratio. The common ratio in this case is 10/11, and the formula is simplified to 11 times one minus (10/11) raised to the 50th power, divided by one over 11. The arithmetic involved is highlighted as the final step in solving the problem.

05:01
📈 Recursive Series and Sum Calculation

The second paragraph discusses a series defined recursively, starting with the first term of 10, and each subsequent term being the previous term multiplied by 9/10. The focus is on finding the sum of the first 30 terms of this series, which is not initially presented as a geometric series. However, it is reformulated using the sum formula for a geometric series, where the common ratio is identified as 9/10. The sum of the first 30 terms is calculated using the formula, which simplifies to 100 times one minus (9/10) raised to the 30th power. Emphasis is placed on the importance of correctly applying parentheses when dealing with powers in the formula.

Mindmap
Keywords
💡Geometric Series
A geometric series 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 the video, the concept is central to finding the sum of the series, with the script discussing how to apply the formula for the sum of a finite geometric series, which is crucial for understanding the calculations presented.
💡Common Ratio
The common ratio in a geometric series is the factor by which we multiply each term to get the next. The script explains that identifying the common ratio is essential for calculating the sum of the series, as seen when the series multiplies by 10/11 or -0.99 to transition from one term to the next.
💡Sum of a Series
The sum of a series refers to the result of adding all the terms of that series together. The video script focuses on finding the sum of the first 50 terms and the first 80 terms of different geometric series, demonstrating the application of formulas to calculate these sums efficiently.
💡First Term
The first term of a series is the initial value from which subsequent terms are generated. In the context of the video, the first term is essential for calculating the sum of a geometric series, as it is multiplied by the common ratio to generate the subsequent terms and is used as the base in the sum formula.
💡Power
In mathematics, a power refers to the result of multiplying a number by itself a certain number of times, denoted as 'b' raised to the power of 'n' (b^n). The script uses the concept of power to describe how the common ratio is raised to the power of the number of terms in the series, which is necessary for the sum formula.
💡Arithmetic
Arithmetic refers to the branch of mathematics dealing with the basic operations of addition, subtraction, multiplication, and division. The script mentions arithmetic when it comes to simplifying the expressions obtained from the sum formula, indicating that once the formula is applied, the remaining steps involve basic arithmetic calculations.
💡Recursively Defined Series
A series defined recursively has each of its terms defined in relation to the previous terms, often through a formula. The script discusses a series where each term is the previous term multiplied by 9/10, illustrating how to calculate the sum of such a series when it is not immediately recognizable as a geometric series.
💡Zeroth Power
The zeroth power of any non-zero number is 1. In the script, the concept is used to clarify that the first term of a series involving powers of 0.99 is actually 1, which is an important distinction when counting the number of terms in the series.
💡Negative Sign
The negative sign in the script is crucial for understanding the common ratio of a series that alternates in sign. It is used to indicate that the common ratio is -0.99, which affects how the terms of the series are calculated and the final sum formula.
💡Even Power
An even power results in a positive number regardless of the base's sign. The script mentions that raising -0.99 to the 80th power results in a positive number because 80 is an even number, which simplifies the calculation of the series sum.
💡Simplification
Simplification in mathematics refers to making a complex expression or equation easier to understand or solve. The script discusses simplifying the sum formula for geometric series by factoring and reducing the expression to a more manageable form, which is necessary for performing the arithmetic calculations.
Highlights

Introduction to finding the sum of the first 50 terms of a geometric series with a common ratio of 10/11.

Explanation of the geometric series formula for the sum of the first n terms.

Application of the formula to calculate the sum, emphasizing the importance of the common ratio.

Simplification of the formula by multiplying the numerator by 11 to clarify the arithmetic process.

Discussion on the importance of recognizing the number of terms in a series, especially when powers are involved.

Clarification that the first term corresponds to the zeroth power in the series involving 0.99.

Identification of the 80th term in the series and the calculation of S sub 80.

Explanation of the common ratio change due to alternating signs in the series.

Demonstration of how to handle negative common ratios in geometric series calculations.

Simplification of the sum formula by considering the even power of a negative number.

Introduction to a series defined recursively and its initial term of 10.

Description of the recursive relationship between terms in the series.

Calculation of the sum of the first 30 terms using the recursive series formula.

Emphasis on the importance of parentheses in mathematical expressions to avoid ambiguity.

Final simplification of the sum formula for the recursive series, highlighting the arithmetic involved.

Transcripts

Browse More Related Video

Geometric Series (Precalculus - College Algebra 72)

Another derivation of the sum of an infinite geometric series | Precalculus | Khan Academy

Sum of an infinite geometric series | Sequences, series and induction | Precalculus | Khan Academy

ILLUSTRATING SEQUENCES AND SERIES || PRECALCULUS

Geometric Sequences (Precalculus - College Algebra 71)

Finite geometric series formula justification | High School Math | Khan Academy

Rate This

5.0 / 5 (0 votes)

Thanks for rating:

Related Tags
Geometric SeriesMathematicsSum CalculationSeries AnalysisCommon RatioPower SeriesEducational ContentArithmetic SimplificationMath TutorialSeries Expansion