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Magazine Chapter 9: Problem 22
Use the formulas in Equation 9.2 to find the sum. $$ \sum_{k=0}^{5} 2\left(\frac{1}{4}\right)^{k} $$
Short Answer
Expert verified
The sum of the series is approximately 2.664.
Step by step solution
01
Understand the Expression
The given expression is a geometric series of the form \( \sum_{k=0}^{n} ar^k \) where \( a = 2 \) and \( r = \frac{1}{4} \). Here, the series ends at \( n = 5 \).
02
Apply the Sum Formula for a Geometric Series
The sum of the first \( n+1 \) terms of a geometric series can be calculated using the formula: \[ S = a \frac{1-r^{(n+1)}}{1-r} \]where \( a \) is the first term and \( r \) is the common ratio.
03
Substitute the Known Values into the Formula
Using the values \( a = 2 \), \( r = \frac{1}{4} \), and \( n = 5 \), substitute them into the sum formula:\[ S = 2 \frac{1-(\frac{1}{4})^{6}}{1-\frac{1}{4}} \]
04
Simplify the Expression
Calculate the powers and simplify:1. Calculate \((\frac{1}{4})^{6} = \frac{1}{4096}\).2. Subtract from one: \(1 - \frac{1}{4096} = \frac{4095}{4096}\).3. Simplify the denominator \(1 - \frac{1}{4} = \frac{3}{4}\).4. Substitute back into the formula:\[ S = 2 \frac{\frac{4095}{4096}}{\frac{3}{4}} \].
05
Solve for the Sum
Continue simplifying the expression:Multiply by the inverse of the denominator: \[ S = 2 \times \frac{4095}{4096} \times \frac{4}{3} \]Calculate the product: \[ S = \frac{8190}{3072} \] Finally, simplify the fraction: \[ S = \frac{2730}{1024} = 2.6640625 \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sum Formula
In mathematics, the sum formula for a geometric series allows us to find the total of all terms in the series. A geometric series is composed of terms that have a constant ratio between them. The sum formula for the first \( n+1 \) terms is:
- \( S \) is the sum of the series.
- \( a \) is the first term of the series.
- \( r \) is the common ratio between consecutive terms.
This formula is useful because it allows us to compute the sum without adding each term individually. By substituting the known values for \( a \), \( r \), and \( n \), we can find the sum efficiently.
- \( S = a \frac{1-r^{(n+1)}}{1-r} \)
- \( S \) is the sum of the series.
- \( a \) is the first term of the series.
- \( r \) is the common ratio between consecutive terms.
This formula is useful because it allows us to compute the sum without adding each term individually. By substituting the known values for \( a \), \( r \), and \( n \), we can find the sum efficiently.
Common Ratio
In a geometric sequence, the common ratio \( r \) is the factor by which each term is multiplied to get the next term. For the series, each term is derived by multiplying the previous term by this constant quotient.
This tells us that each subsequent term is a quarter of the preceding one.
Understanding the common ratio is crucial for determining the behavior and sum of the series.
- If \( r > 1 \), the terms will grow larger (diverging series).
- If \( r < 1 \), the terms get progressively smaller (converging series).
- If \( r = 1 \), all terms in the sequence are the same.
This tells us that each subsequent term is a quarter of the preceding one.
Understanding the common ratio is crucial for determining the behavior and sum of the series.
First Term
The first term, often denoted as \( a \), is the starting point of a geometric sequence. It plays a crucial role in calculating the total sum of a geometric series. It sets the initial value from which all subsequent terms are calculated by continuously multiplying by the common ratio.
Starting from this first term, we multiply by the common ratio \( \frac{1}{4} \) to generate further terms.
Remember, knowing the first term is essential for predicting and summing the sequence effectively.
- The first term \( a \) is also used directly in the sum formula \( S = a \frac{1-r^{(n+1)}}{1-r} \).
Starting from this first term, we multiply by the common ratio \( \frac{1}{4} \) to generate further terms.
Remember, knowing the first term is essential for predicting and summing the sequence effectively.
Geometric Sequence
A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. These sequences can be finite or infinite:
The series can be explicitly mentioned in summation notation, like the given example \( \sum_{k=0}^{5} 2\left(\frac{1}{4}\right)^{k} \) which sums all terms starting from \( k=0 \) to \( k=5 \).
Recognizing and working with geometric sequences is essential in various mathematical applications and problem-solving scenarios.
- Finite geometric sequences end after a limited number of terms.
- Infinite geometric sequences continue indefinitely.
The series can be explicitly mentioned in summation notation, like the given example \( \sum_{k=0}^{5} 2\left(\frac{1}{4}\right)^{k} \) which sums all terms starting from \( k=0 \) to \( k=5 \).
Recognizing and working with geometric sequences is essential in various mathematical applications and problem-solving scenarios.
