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Answer :
To find the first five terms of the sequence given by the formula [tex]\( b_n = 400 \cdot \left(-\frac{1}{2}\right)^{n-1} \)[/tex], we'll evaluate this expression for the first five positive integer values of [tex]\( n \)[/tex].
Step-by-step calculation:
1. First Term ([tex]\( n = 1 \)[/tex]):
[tex]\[
b_1 = 400 \cdot \left(-\frac{1}{2}\right)^{1-1} = 400 \cdot \left(-\frac{1}{2}\right)^0 = 400 \cdot 1 = 400
\][/tex]
2. Second Term ([tex]\( n = 2 \)[/tex]):
[tex]\[
b_2 = 400 \cdot \left(-\frac{1}{2}\right)^{2-1} = 400 \cdot \left(-\frac{1}{2}\right)^1 = 400 \cdot \left(-\frac{1}{2}\right) = -200
\][/tex]
3. Third Term ([tex]\( n = 3 \)[/tex]):
[tex]\[
b_3 = 400 \cdot \left(-\frac{1}{2}\right)^{3-1} = 400 \cdot \left(-\frac{1}{2}\right)^2 = 400 \cdot \frac{1}{4} = 100
\][/tex]
4. Fourth Term ([tex]\( n = 4 \)[/tex]):
[tex]\[
b_4 = 400 \cdot \left(-\frac{1}{2}\right)^{4-1} = 400 \cdot \left(-\frac{1}{2}\right)^3 = 400 \cdot \left(-\frac{1}{8}\right) = -50
\][/tex]
5. Fifth Term ([tex]\( n = 5 \)[/tex]):
[tex]\[
b_5 = 400 \cdot \left(-\frac{1}{2}\right)^{5-1} = 400 \cdot \left(-\frac{1}{2}\right)^4 = 400 \cdot \frac{1}{16} = 25
\][/tex]
So, the first five terms of the sequence are [tex]\( 400, -200, 100, -50, \)[/tex] and [tex]\( 25 \)[/tex].
The correct answer is C. [tex]\( 400, -200, 100, -50, 25 \)[/tex].
Step-by-step calculation:
1. First Term ([tex]\( n = 1 \)[/tex]):
[tex]\[
b_1 = 400 \cdot \left(-\frac{1}{2}\right)^{1-1} = 400 \cdot \left(-\frac{1}{2}\right)^0 = 400 \cdot 1 = 400
\][/tex]
2. Second Term ([tex]\( n = 2 \)[/tex]):
[tex]\[
b_2 = 400 \cdot \left(-\frac{1}{2}\right)^{2-1} = 400 \cdot \left(-\frac{1}{2}\right)^1 = 400 \cdot \left(-\frac{1}{2}\right) = -200
\][/tex]
3. Third Term ([tex]\( n = 3 \)[/tex]):
[tex]\[
b_3 = 400 \cdot \left(-\frac{1}{2}\right)^{3-1} = 400 \cdot \left(-\frac{1}{2}\right)^2 = 400 \cdot \frac{1}{4} = 100
\][/tex]
4. Fourth Term ([tex]\( n = 4 \)[/tex]):
[tex]\[
b_4 = 400 \cdot \left(-\frac{1}{2}\right)^{4-1} = 400 \cdot \left(-\frac{1}{2}\right)^3 = 400 \cdot \left(-\frac{1}{8}\right) = -50
\][/tex]
5. Fifth Term ([tex]\( n = 5 \)[/tex]):
[tex]\[
b_5 = 400 \cdot \left(-\frac{1}{2}\right)^{5-1} = 400 \cdot \left(-\frac{1}{2}\right)^4 = 400 \cdot \frac{1}{16} = 25
\][/tex]
So, the first five terms of the sequence are [tex]\( 400, -200, 100, -50, \)[/tex] and [tex]\( 25 \)[/tex].
The correct answer is C. [tex]\( 400, -200, 100, -50, 25 \)[/tex].
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