To find the first five terms of the geometric sequence where \( a_1 = 64 \) and the common ratio \( r = \frac{5}{4} \), we can use the formula for the nth term of a geometric sequence, which is given by:
\[ a_n = a_1 \cdot r^{(n-1)} \]
Let's calculate the first five terms:
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First term (\( a_1 \)): \[ a_1 = 64 \]
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Second term (\( a_2 \)): \[ a_2 = a_1 \cdot r = 64 \cdot \frac{5}{4} = 64 \cdot 1.25 = 80 \]
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Third term (\( a_3 \)): \[ a_3 = a_1 \cdot r^2 = 64 \cdot \left(\frac{5}{4}\right)^2 = 64 \cdot \frac{25}{16} = 64 \cdot 1.5625 = 100 \]
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Fourth term (\( a_4 \)): \[ a_4 = a_1 \cdot r^3 = 64 \cdot \left(\frac{5}{4}\right)^3 = 64 \cdot \frac{125}{64} = 125 \]
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Fifth term (\( a_5 \)): \[ a_5 = a_1 \cdot r^4 = 64 \cdot \left(\frac{5}{4}\right)^4 = 64 \cdot \frac{625}{256} = \frac{40000}{256} = \frac{625}{4} \]
Therefore, the first five terms of the geometric sequence are:
\[ 64, 80, 100, 125, \frac{625}{4} \]
The corresponding response is:
64, 80, 100, 125, \( \frac{625}{4} \).