To determine which expression provides the same result as [tex]\(\sum_{i=0}^4\left(5 \left(\frac{1}{3}\right)^i\right)\)[/tex], we need to evaluate each given option and compare the results.
1. Option A:
[tex]\[
5 \sum_{i=0}^1 \left(\frac{1}{3}\right)^i
\][/tex]
This option involves summing the series [tex]\(\left(\frac{1}{3}\right)^i\)[/tex] for [tex]\(i = 0\)[/tex] to [tex]\(1\)[/tex], then multiplying by [tex]\(5\)[/tex]. Evaluating this:
[tex]\[
5 \left( \left(\frac{1}{3}\right)^0 + \left(\frac{1}{3}\right)^1 \right) = 5 (1 + \frac{1}{3}) = 5 \times \frac{4}{3} = \frac{20}{3} \approx 6.67
\][/tex]
The result is approximately [tex]\(6.67\)[/tex].
2. Option B:
[tex]\[
\left(\frac{1}{3}\right) \sum_{i=0}^4 5^i
\][/tex]
This option involves summing the series [tex]\(5^i\)[/tex] for [tex]\(i = 0\)[/tex] to [tex]\(4\)[/tex], then multiplying by [tex]\(\frac{1}{3}\)[/tex]. Evaluating this:
[tex]\[
\sum_{i=0}^4 5^i = 5^0 + 5^1 + 5^2 + 5^3 + 5^4 = 1 + 5 + 25 + 125 + 625 = 781
\][/tex]
Then,
[tex]\[
\left(\frac{1}{3}\right) \times 781 = \frac{781}{3} \approx 260.33
\][/tex]
The result is approximately [tex]\(260.33\)[/tex].
3. Option C:
[tex]\[
\sum_{i=0}^4 \left(5 \left(\frac{1}{3}\right)^i\right)
\][/tex]
This is the original series given in the problem. Evaluating this:
[tex]\[
5 \left(\frac{1}{3}\right)^0 + 5 \left(\frac{1}{3}\right)^1 + 5 \left(\frac{1}{3}\right)^2 + 5 \left(\frac{1}{3}\right)^3 + 5 \left(\frac{1}{3}\right)^4
\][/tex]
Simplifying each term:
[tex]\[
5 \cdot 1 + 5 \cdot \frac{1}{3} + 5 \cdot \left(\frac{1}{3}\right)^2 + 5 \cdot \left(\frac{1}{3}\right)^3 + 5 \cdot \left(\frac{1}{3}\right)^4
\][/tex]
[tex]\[
= 5 + \frac{5}{3} + \frac{5}{9} + \frac{5}{27} + \frac{5}{81}
\][/tex]
Adding these fractions:
[tex]\[
\approx 5 + 1.67 + 0.56 + 0.19 + 0.06 = 7.469
\][/tex]
The result is approximately [tex]\(7.47\)[/tex].
4. Option D:
[tex]\[
\left(\frac{1}{3}\right)^4 \sum_{i=0}^4 5
\][/tex]
This option involves summing [tex]\(\sum_{i=0}^4 5\)[/tex], then multiplying by [tex]\(\left(\frac{1}{3}\right)^4\)[/tex]. Evaluating this:
[tex]\[
\sum_{i=0}^4 5 = 5 + 5 + 5 + 5 + 5 = 25
\][/tex]
Then,
[tex]\[
\left(\frac{1}{3}\right)^4 \times 25 = \frac{1}{81} \times 25 = \frac{25}{81} \approx 0.31
\][/tex]
The result is approximately [tex]\(0.31\)[/tex].
Comparing the results:
- Option A: [tex]\(6.67\)[/tex]
- Option B: [tex]\(260.33\)[/tex]
- Option C: [tex]\(7.47\)[/tex]
- Option D: [tex]\(0.31\)[/tex]
The correct answer matches the original series' result, which is approximately [tex]\(7.47\)[/tex]. Therefore, the correct answer is:
C. [tex]\(\sum_{i=0}^4 \left(5 \left(\frac{1}{3}\right)^i\right)\)[/tex]
