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Mobile App Chapter 19: Problem 6
A geometric series has \(S_{3}=\frac{37}{8}\) and \(S_{6}=\frac{3367}{512}\). Find the first term and the common ratio.
Short Answer
Expert verified
Answer: The first term (a) is 37/8 and the common ratio (r) is 1.
Step by step solution
01
Writing equations for given values
We have the formula as \(S_n = a\frac{1 - r^n}{1 - r}\). We know \(S_3 = \frac{37}{8}\) and \(S_6 =\frac{3367}{512}\), so the equations are:
1) \(a\frac{1 - r^3}{1 - r} = \frac{37}{8}\)
2) \(a\frac{1 - r^6}{1 - r} = \frac{3367}{512}\)
02
Divide equation 1 by equation 2
Divide equation 1 by equation 2 to eliminate \(a\):
\(\frac{1 - r^3}{1 - r^6} = \frac{\frac{37}{8}}{\frac{3367}{512}}\)
Simplify the equation:
\(\frac{1 - r^3}{1 - r^6} = \frac{37}{8} \cdot \frac{512}{3367}\)
03
Solve for \(r\)
Now, let's find \(r\) by solving the equation:
\(\frac{1 - r^3}{1 - r^6} = \frac{37 \cdot 512}{8 \cdot 3367}\)
\(1 - r^3 = \frac{37 \cdot 512}{8 \cdot 3367}(1 - r^6)\)
\(r^6 - r^3 = \frac{37 \cdot 512}{8 \cdot 3367}r^6 - \frac{37 \cdot 512}{8 \cdot 3367}r^3\)
Putting everything on the left-hand side:
\(r^3(\frac{37 \cdot 512}{8 \cdot 3367} - 1) = r^6(\frac{37 \cdot 512}{8 \cdot 3367} - 1)\)
As both terms share a common factor, let's divide by \((\frac{37 \cdot 512}{8 \cdot 3367} - 1)\):
\(r^3 = r^6\)
This means,
\(r^3 = 1\)
Now, we can find out the possible values for \(r\) by taking the cube root of both sides:
\(r = \sqrt[3]{1}\)
\(r = 1\) (We reject this value as this would mean the series is not geometric)
So, we have only one possible value for \(r\):
\(r = 1\)
04
Solve for \(a\) using either equation
Now, we will use the first equation to find the value of \(a\).
\(a\frac{1 - r^3}{1 - r} = \frac{37}{8}\)
Plug in the value of \(r\):
\(a\frac{1 - 1}{1 - 1} = \frac{37}{8}\)
This tells us that the first term, \(a\), is equal to \(\frac{37}{8}\).
\(a = \frac{37}{8}\).
05
Write the final answer
In conclusion, we found that the first term \(a\) is \(\frac{37}{8}\) and the common ratio \(r\) is \(1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sum of Geometric Series
Understanding the sum of a geometric series is crucial when working with sequences in mathematics. 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.
The sum of the first 'n' terms of a geometric series can be calculated using the formula: \[ S_n = a\frac{1 - r^n}{1 - r} \]
where:
When the common ratio \( r \) is greater than 1, the terms of the series grow larger, and when \( r \) is between -1 and 1, the terms decrease in magnitude. The formula is especially potent for finite geometric series and is crucial for solving problems where the sum of several terms of a series is known, like in the given exercise.
The sum of the first 'n' terms of a geometric series can be calculated using the formula: \[ S_n = a\frac{1 - r^n}{1 - r} \]
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.
When the common ratio \( r \) is greater than 1, the terms of the series grow larger, and when \( r \) is between -1 and 1, the terms decrease in magnitude. The formula is especially potent for finite geometric series and is crucial for solving problems where the sum of several terms of a series is known, like in the given exercise.
Common Ratio
The common ratio is a key determinant of a geometric series and is denoted as \( r \). It is found by dividing any term in the series by the preceding term. This ratio remains constant between consecutive terms throughout the series.
For example, in a series where the second term is twice as large as the first, the common ratio is 2. This ratio is not only relevant in identifying the type of series but also in determining the behavior of the series over time. Series with a common ratio greater than 1 will increase endlessly, while a ratio between 0 and 1 signifies a series that approaches zero as it progresses. Negative common ratios will result in terms alternating in sign.
Crucially, in solving problems like the textbook exercise, identifying the common ratio requires an understanding of how to manipulate equations and use algebraic expressions to reveal its value.
For example, in a series where the second term is twice as large as the first, the common ratio is 2. This ratio is not only relevant in identifying the type of series but also in determining the behavior of the series over time. Series with a common ratio greater than 1 will increase endlessly, while a ratio between 0 and 1 signifies a series that approaches zero as it progresses. Negative common ratios will result in terms alternating in sign.
Crucially, in solving problems like the textbook exercise, identifying the common ratio requires an understanding of how to manipulate equations and use algebraic expressions to reveal its value.
First Term of Geometric Series
The first term of a geometric series, represented by \( a \), sets the stage for the entire series. It is the starting point from which all other terms are derived through multiplication by the common ratio \( r \).
Knowing the first term is essential for calculating any further elements of the series or the sum of the series. The given exercise demonstrates finding the first term when the sums of a certain number of terms are provided. By cleverly manipulating the formulas and using the values of sums provided, one can extract valuable information about the first term and hence, define the specific geometric series in question.
Finding the correct value of \( a \) when the common ratio is known, involves substituting back into the sum formula, leading us to the starting point of the sequence, which in turn unlocks the ability to express any term of the series or its sum.
Knowing the first term is essential for calculating any further elements of the series or the sum of the series. The given exercise demonstrates finding the first term when the sums of a certain number of terms are provided. By cleverly manipulating the formulas and using the values of sums provided, one can extract valuable information about the first term and hence, define the specific geometric series in question.
Finding the correct value of \( a \) when the common ratio is known, involves substituting back into the sum formula, leading us to the starting point of the sequence, which in turn unlocks the ability to express any term of the series or its sum.
