Problem 5 About sums \(f_{j}\) and differe... [FREE SOLUTION]

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Chapter 5: Problem 5

About sums \(f_{j}\) and differences \(\boldsymbol{v}_{j}\). Show that \(f_{j}=r_{i}^{j}(r-1)\) has \(f_{j}-f_{j-1}=r^{j}\) '. Therefore the geometric series \(1+r+\cdots+r^{j-1}\) adds up to _____ (remember to subtract \(f_{0}\) ).

Short Answer

Expert verified

The series sums to \( \frac{r^j - 1}{r - 1} \).

Step by step solution

Recap the Formula

We are given the formula for the function as \( f_j = r^j (r-1) \). We want to show that \( f_j - f_{j-1} = r^j \).

Substitute and Expand

Substitute \( j-1 \) in the formula to get \( f_{j-1} = r^{j-1} (r-1) \). Then, calculate \( f_j - f_{j-1} \).

Calculate the Difference

Compute \( f_j - f_{j-1} = r^j (r-1) - r^{j-1} (r-1) \). This simplifies to \((r^j - r^{j-1})(r-1) \).

Factor Out the Common Term

Factor out \( r^{j-1} \) from the term \((r^j - r^{j-1})\) to get \( r^{j-1} (r - 1) \). Then, multiply by \( (r-1) \, \) giving \( r^j \).

Derive the Geometric Series Sum Formula

Recognize that \( f_j = r^j (r-1) \) arose from the expression for the sum of a geometric series. Using the result, the sum of the series \( 1 + r + \cdots + r^{j-1} \) is \( \frac{r^j - 1}{r - 1} \) after subtracting \( f_0 = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Geometric Series

A geometric series is a sum of terms where each term is a constant multiple, known as the ratio, of the previous term. To find the sum of a geometric series, such as \( 1 + r + r^2 + \, \cdots + \, r^{j-1} \), you can use a handy formula. This series can be expressed in a compact, formulaic way: \( \frac{r^j - 1}{r - 1} \).
For clarity:

\( n \) is the number of terms.
\( r \) is the common ratio.
\( 1 \) is the first term.

Imagine you have to sum the first \( j \) terms of this series. The formula saves you from tedious calculations, giving you a result quickly. This sum formula is crucial in various mathematical applications, such as calculating interest or predicting growth patterns.

Formula Derivation

Deriving the formula for the sum of a geometric series involves a little algebraic manipulation. Let's see how it's done. Begin with the sum:

\( S = 1 + r + r^2 + \, \cdots + \, r^{j-1} \)

Then, multiply the entire sum \( S \) by the common ratio \( r \):

\( rS = r + r^2 + r^3 + \, \cdots + \, r^j \)

Now, subtract the second equation from the first:

\( S - rS = 1 - r^j \)

Factor out \( S \) to get:

\( S (1 - r) = 1 - r^j \)

And solving for \( S \) gives:

\( S = \frac{1 - r^j}{1 - r} \)

This is your formula for the sum of a geometric series with the first term as \( 1 \) and the ratio as \( r \). It elegantly wraps up all terms into a neat expression.

Difference Calculation

Calculating the difference between consecutive terms in a geometric series is often necessary to understand the series' progression. Let's illustrate why this is important using the given functions, \( f_j = r^j (r-1) \) and \( f_{j-1} = r^{j-1} (r-1) \).
To find the difference, subtract \( f_{j-1} \) from \( f_j \):

\( f_j - f_{j-1} = r^j (r-1) - r^{j-1} (r-1) \)

This simplifies to:

\( f_j - f_{j-1} = (r^j - r^{j-1}) (r-1) \)

Factoring out \( r^{j-1} \), we find:

\( r^{j-1}(r-1)(r-1) = r^j - r^{j-1} = r^j (r-1) \)

Eventually, simplifying expression leads to \( r^j \). This process shows that the difference in terms is consistent and adheres to a pattern, underscoring the series' stability.

Sequence and Series

In mathematics, understanding the interplay between sequences and series helps grasp how numbers pattern and accumulate. A sequence is an ordered list of numbers, while a series is the sum of a sequence's terms. Geometric sequences, where each term is a constant multiple of the previous, often morph into geometric series when summed.
Key concepts to remember include:

In a geometric sequence, each term varies by a consistent ratio \( r \).
A geometric series is formed by summing the elements of a geometric sequence.
The geometric series sum formula is \( \frac{r^j - 1}{r - 1} \), applicable for any \( j \) terms.

By recognizing patterns in sequences, and using series formulas, mathematicians can solve complex problems swiftly. These concepts build understanding of important mathematical behavior and relationships.

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91影视

About sums \(f_{j}\) and differences \(\boldsymbol{v}_{j}\). Show that \(f_{j}=r_{i}^{j}(r-1)\) has \(f_{j}-f_{j-1}=r^{j}\) '. Therefore the geometric series \(1+r+\cdots+r^{j-1}\) adds up to _____ (remember to subtract \(f_{0}\) ).

Short Answer

Step by step solution

Recap the Formula

Substitute and Expand

Calculate the Difference

Factor Out the Common Term

Derive the Geometric Series Sum Formula

Key Concepts

Sum of Geometric Series

Formula Derivation

Difference Calculation

Sequence and Series

One App. One Place for Learning.

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