Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To determine which expression gives the same result as [tex]\(\sum_{i=0}^4 \left(5 \left(\frac{1}{3} \right)^i \right)\)[/tex], let's break down the given summation and analyze the geometric series involved.
1. Understanding the Series:
The given series is [tex]\(\sum_{i=0}^4 \left(5 \left(\frac{1}{3}\right)^i \right)\)[/tex].
- This is a geometric series where each term is of the form [tex]\(a \cdot r^i\)[/tex].
- Here, [tex]\(a=5\)[/tex] (the first term) and [tex]\(r=\frac{1}{3}\)[/tex] (the common ratio).
2. Sum of a Geometric Series:
For a geometric series of the form [tex]\(a + ar + ar^2 + \ldots + ar^{n-1}\)[/tex], the sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms is given by:
[tex]\[ S_n = \frac{a(1 - r^n)}{1 - r} \][/tex]
3. Calculation with Given Values:
- In the given series, [tex]\(a=5\)[/tex], [tex]\(r =\frac{1}{3}\)[/tex], and there are 5 terms (from [tex]\(i=0\)[/tex] to [tex]\(i=4\)[/tex]).
- Applying the formula for the sum of the geometric series:
[tex]\[ S_5 = 5 \cdot \frac{1-(\frac{1}{3})^5}{1 - \frac{1}{3}} \][/tex]
- Simplifying the denominator:
[tex]\[ 1 - \frac{1}{3} = \frac{2}{3} \][/tex]
- So, the sum [tex]\(S_5\)[/tex] becomes:
[tex]\[ S_5 = 5 \cdot \frac{1 - \left(\frac{1}{3}\right)^5}{\frac{2}{3}} = 5 \cdot \frac{3}{2} \left(1 - \left(\frac{1}{3}\right)^5 \right) = \frac{15}{2} \left(1 - \left(\frac{1}{3}\right)^5 \right) \][/tex]
4. Numerical Result:
- Compute [tex]\(\left(\frac{1}{3}\right)^5 = \frac{1}{243}\)[/tex]
- Hence:
[tex]\[ 1 - \frac{1}{243} = \frac{242}{243} \][/tex]
- Now, substituting back:
[tex]\[ S_5 = \frac{15}{2} \cdot \frac{242}{243} = \frac{15 \cdot 242}{2 \cdot 243} = \frac{3630}{486} = 7.469135802469135 \][/tex]
5. Identifying the Correct Expression:
Option D:
[tex]\[ 5 \sum_{i=0}^4 \left(\frac{1}{3}\right)^i \][/tex]
This can be rewritten using the sum formula for a geometric series:
[tex]\[ 5 \cdot \frac{1 - \left(\frac{1}{3}\right)^5}{1 - \frac{1}{3}} = 5 \cdot \frac{1 - \frac{1}{243}}{\frac{2}{3}} = 5 \cdot \frac{243 - 1}{243} \cdot \frac{3}{2} = 5 \cdot \frac{242}{243} \cdot \frac{3}{2} = \frac{15 \cdot 242}{2 \cdot 243} = 7.469135802469135 \][/tex]
Therefore, the expression that gives the same result as [tex]\(\sum_{i=0}^4 \left(5 \left(\frac{1}{3} \right)^i\right)\)[/tex] is:
\[
\boxed{D}
1. Understanding the Series:
The given series is [tex]\(\sum_{i=0}^4 \left(5 \left(\frac{1}{3}\right)^i \right)\)[/tex].
- This is a geometric series where each term is of the form [tex]\(a \cdot r^i\)[/tex].
- Here, [tex]\(a=5\)[/tex] (the first term) and [tex]\(r=\frac{1}{3}\)[/tex] (the common ratio).
2. Sum of a Geometric Series:
For a geometric series of the form [tex]\(a + ar + ar^2 + \ldots + ar^{n-1}\)[/tex], the sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms is given by:
[tex]\[ S_n = \frac{a(1 - r^n)}{1 - r} \][/tex]
3. Calculation with Given Values:
- In the given series, [tex]\(a=5\)[/tex], [tex]\(r =\frac{1}{3}\)[/tex], and there are 5 terms (from [tex]\(i=0\)[/tex] to [tex]\(i=4\)[/tex]).
- Applying the formula for the sum of the geometric series:
[tex]\[ S_5 = 5 \cdot \frac{1-(\frac{1}{3})^5}{1 - \frac{1}{3}} \][/tex]
- Simplifying the denominator:
[tex]\[ 1 - \frac{1}{3} = \frac{2}{3} \][/tex]
- So, the sum [tex]\(S_5\)[/tex] becomes:
[tex]\[ S_5 = 5 \cdot \frac{1 - \left(\frac{1}{3}\right)^5}{\frac{2}{3}} = 5 \cdot \frac{3}{2} \left(1 - \left(\frac{1}{3}\right)^5 \right) = \frac{15}{2} \left(1 - \left(\frac{1}{3}\right)^5 \right) \][/tex]
4. Numerical Result:
- Compute [tex]\(\left(\frac{1}{3}\right)^5 = \frac{1}{243}\)[/tex]
- Hence:
[tex]\[ 1 - \frac{1}{243} = \frac{242}{243} \][/tex]
- Now, substituting back:
[tex]\[ S_5 = \frac{15}{2} \cdot \frac{242}{243} = \frac{15 \cdot 242}{2 \cdot 243} = \frac{3630}{486} = 7.469135802469135 \][/tex]
5. Identifying the Correct Expression:
Option D:
[tex]\[ 5 \sum_{i=0}^4 \left(\frac{1}{3}\right)^i \][/tex]
This can be rewritten using the sum formula for a geometric series:
[tex]\[ 5 \cdot \frac{1 - \left(\frac{1}{3}\right)^5}{1 - \frac{1}{3}} = 5 \cdot \frac{1 - \frac{1}{243}}{\frac{2}{3}} = 5 \cdot \frac{243 - 1}{243} \cdot \frac{3}{2} = 5 \cdot \frac{242}{243} \cdot \frac{3}{2} = \frac{15 \cdot 242}{2 \cdot 243} = 7.469135802469135 \][/tex]
Therefore, the expression that gives the same result as [tex]\(\sum_{i=0}^4 \left(5 \left(\frac{1}{3} \right)^i\right)\)[/tex] is:
\[
\boxed{D}
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
