To find the first five terms of the geometric sequence with the first term \( a_1 = -2 \) and common ratio \( r = -\frac{5}{2} \), we can use the formula for the \( n \)-th term of a geometric sequence:
\[ a_n = a_1 \cdot r^{(n-1)} \]
Now, let's calculate the first five terms:
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First term (\( n = 1 \)): \[ a_1 = -2 \]
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Second term (\( n = 2 \)): \[ a_2 = a_1 \cdot r^{(2-1)} = -2 \cdot \left(-\frac{5}{2}\right) = -2 \cdot -\frac{5}{2} = 5 \]
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Third term (\( n = 3 \)): \[ a_3 = a_1 \cdot r^{(3-1)} = -2 \cdot \left(-\frac{5}{2}\right)^2 = -2 \cdot \frac{25}{4} = -\frac{50}{4} = -\frac{25}{2} \]
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Fourth term (\( n = 4 \)): \[ a_4 = a_1 \cdot r^{(4-1)} = -2 \cdot \left(-\frac{5}{2}\right)^3 = -2 \cdot -\frac{125}{8} = \frac{250}{8} = \frac{125}{4} \]
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Fifth term (\( n = 5 \)): \[ a_5 = a_1 \cdot r^{(5-1)} = -2 \cdot \left(-\frac{5}{2}\right)^4 = -2 \cdot \frac{625}{16} = -\frac{1250}{16} = -\frac{625}{8} \]
Putting it all together, the first five terms of the geometric sequence are:
\[ \left{ a_n \right} = \left{ -2, 5, -\frac{25}{2}, \frac{125}{4}, -\frac{625}{8} \right} \]
However, to match the format of the provided options, here are the terms we calculated:
- \( -2 \)
- \( 5 \)
- \( -\frac{25}{2} \)
- \( \frac{125}{4} \)
- \( -\frac{625}{8} \)
Among the options given in your question, the closest option capturing the pattern of these terms is:
left-brace ( a_n \right} equals left-brace -2, 5, -\frac{25}{2}, \frac{125}{4}, \ldots right-brace