Answer :
To find the approximate value of the sum [tex]\(\sum_{n=1}^5 3(0.2)^{n-1}\)[/tex], let's break it down step-by-step.
### Step-by-Step Calculation:
We are given the sum [tex]\(\sum_{n=1}^5 3(0.2)^{n-1}\)[/tex].
1. Calculate individual terms:
- For [tex]\(n = 1\)[/tex]:
[tex]\[ 3(0.2)^{1-1} = 3(0.2)^0 = 3 \cdot 1 = 3 \][/tex]
- For [tex]\(n = 2\)[/tex]:
[tex]\[ 3(0.2)^{2-1} = 3(0.2)^1 = 3 \cdot 0.2 = 0.6 \][/tex]
- For [tex]\(n = 3\)[/tex]:
[tex]\[ 3(0.2)^{3-1} = 3(0.2)^2 = 3 \cdot 0.04 = 0.12 \][/tex]
- For [tex]\(n = 4\)[/tex]:
[tex]\[ 3(0.2)^{4-1} = 3(0.2)^3 = 3 \cdot 0.008 = 0.024 \][/tex]
- For [tex]\(n = 5\)[/tex]:
[tex]\[ 3(0.2)^{5-1} = 3(0.2)^4 = 3 \cdot 0.0016 = 0.0048 \][/tex]
2. Add all the terms together:
[tex]\[ 3 + 0.6 + 0.12 + 0.024 + 0.0048 = 3.7488 \][/tex]
3. Round to the nearest hundredth:
[tex]\[ 3.7488 \approx 3.75 \][/tex]
Thus, the approximate value of the sum is [tex]\(\boxed{3.75}\)[/tex].
### Answer Choices Evaluation:
- 1.25
- 3.00
- 3.75
- 4.25
The correct option, after rounding to the nearest hundredth, is [tex]\(\boxed{3.75}\)[/tex].
### Step-by-Step Calculation:
We are given the sum [tex]\(\sum_{n=1}^5 3(0.2)^{n-1}\)[/tex].
1. Calculate individual terms:
- For [tex]\(n = 1\)[/tex]:
[tex]\[ 3(0.2)^{1-1} = 3(0.2)^0 = 3 \cdot 1 = 3 \][/tex]
- For [tex]\(n = 2\)[/tex]:
[tex]\[ 3(0.2)^{2-1} = 3(0.2)^1 = 3 \cdot 0.2 = 0.6 \][/tex]
- For [tex]\(n = 3\)[/tex]:
[tex]\[ 3(0.2)^{3-1} = 3(0.2)^2 = 3 \cdot 0.04 = 0.12 \][/tex]
- For [tex]\(n = 4\)[/tex]:
[tex]\[ 3(0.2)^{4-1} = 3(0.2)^3 = 3 \cdot 0.008 = 0.024 \][/tex]
- For [tex]\(n = 5\)[/tex]:
[tex]\[ 3(0.2)^{5-1} = 3(0.2)^4 = 3 \cdot 0.0016 = 0.0048 \][/tex]
2. Add all the terms together:
[tex]\[ 3 + 0.6 + 0.12 + 0.024 + 0.0048 = 3.7488 \][/tex]
3. Round to the nearest hundredth:
[tex]\[ 3.7488 \approx 3.75 \][/tex]
Thus, the approximate value of the sum is [tex]\(\boxed{3.75}\)[/tex].
### Answer Choices Evaluation:
- 1.25
- 3.00
- 3.75
- 4.25
The correct option, after rounding to the nearest hundredth, is [tex]\(\boxed{3.75}\)[/tex].
