Answer :
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}
