Write the first five terms of the geometric sequence with a subscript 1 baseline equals negative 2 and common ratio r equals negative Start Fraction 5 over 2 End Fraction.

Question

Write the first five terms of the geometric sequence with a subscript 1 baseline equals negative 2 and common ratio r equals negative Start Fraction 5 over 2 End Fraction.
(1 point)
Responses

left-brace a subscript n baseline right-brace equals left-brace negative 2 comma negative Start Fraction 9 over 2 End Fraction comma negative 7 comma negative Start Fraction 19 over 2 End Fraction comma ellipsis right-brace
Image with alt text: left-brace a subscript n baseline right-brace equals left-brace negative 2 comma negative Start Fraction 9 over 2 End Fraction comma negative 7 comma negative Start Fraction 19 over 2 End Fraction comma ellipsis right-brace

left-brace a subscript n baseline right-brace equals left-brace negative 2 comma negative 5 comma negative Start Fraction 25 over 2 End Fraction comma negative Start Fraction 125 over 4 End Fraction comma ellipsis right-brace
Image with alt text: left-brace a subscript n baseline right-brace equals left-brace negative 2 comma negative 5 comma negative Start Fraction 25 over 2 End Fraction comma negative Start Fraction 125 over 4 End Fraction comma ellipsis right-brace

left-brace a subscript n baseline right-brace equals left-brace 2 comma negative 5 comma Start Fraction 25 over 2 End Fraction comma negative Start Fraction 125 over 4 End Fraction comma ellipsis right-brace
Image with alt text: left-brace a subscript n baseline right-brace equals left-brace 2 comma negative 5 comma Start Fraction 25 over 2 End Fraction comma negative Start Fraction 125 over 4 End Fraction comma ellipsis right-brace

left-brace a subscript n baseline right-brace equals left-brace negative 2 comma 5 comma negative Start Fraction 25 over 2 End Fraction comma Start Fraction 125 over 4 End Fraction comma ellipsis right-brace

GPT-4o mini · Answer

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:

First term ( $n = 1$ ): $a_{1} = - 2$
Second term ( $n = 2$ ): $a_{2} = a_{1} \cdot r^{(2 - 1)} = - 2 \cdot (- \frac{5}{2}) = - 2 \cdot - \frac{5}{2} = 5$
Third term ( $n = 3$ ): $a_{3} = a_{1} \cdot r^{(3 - 1)} = - 2 \cdot {(- \frac{5}{2})}^{2} = - 2 \cdot \frac{25}{4} = - \frac{50}{4} = - \frac{25}{2}$
Fourth term ( $n = 4$ ): $a_{4} = a_{1} \cdot r^{(4 - 1)} = - 2 \cdot {(- \frac{5}{2})}^{3} = - 2 \cdot - \frac{125}{8} = \frac{250}{8} = \frac{125}{4}$
Fifth term ( $n = 5$ ): $a_{5} = a_{1} \cdot r^{(5 - 1)} = - 2 \cdot {(- \frac{5}{2})}^{4} = - 2 \cdot \frac{625}{16} = - \frac{1250}{16} = - \frac{625}{8}$

Putting it all together, the first five terms of the geometric sequence are:

$Missing or unrecognized delimiter for \left$

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