Answer :
To find the missing terms in a geometric sequence, we need to follow these steps:
1. Identify the Known Terms:
We know the first term ([tex]\(a_1\)[/tex]) is 3125 and the third term ([tex]\(a_3\)[/tex]) is -1024.
2. Understand the Relationship in a Geometric Sequence:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio ([tex]\(r\)[/tex]). Given this relationship, the third term ([tex]\(a_3\)[/tex]) can be expressed as:
[tex]\[
a_3 = a_1 \cdot r^2
\][/tex]
3. Set Up the Equation:
Plug in the known values:
[tex]\[
-1024 = 3125 \cdot r^2
\][/tex]
4. Solve for the Common Ratio ([tex]\(r\)[/tex]):
We can solve for [tex]\(r^2\)[/tex]:
[tex]\[
r^2 = \frac{-1024}{3125}
\][/tex]
Simplifying it, we get:
[tex]\[
r^2 = -0.32768
\][/tex]
Solving for [tex]\(r\)[/tex], we take the square root of both sides:
[tex]\[
r = \sqrt{-0.32768}
\][/tex]
The result involves both real and imaginary parts, so we get:
[tex]\[
r \approx 3.5051436688908974 \times 10^{-17} + 0.5724334022399462j
\][/tex]
5. Find the Second Term ([tex]\(a_2\)[/tex]):
The second term ([tex]\(a_2\)[/tex]) is the first term multiplied by the common ratio [tex]\(r\)[/tex]:
[tex]\[
a_2 = a_1 \cdot r
\][/tex]
Substitute the values:
[tex]\[
a_2 = 3125 \cdot (3.5051436688908974 \times 10^{-17} + 0.5724334022399462j)
\][/tex]
Simplifying, we get:
[tex]\[
a_2 \approx 1.0953573965284054 \times 10^{-13} + 1788.854381999832j
\][/tex]
Therefore, the possible values for the missing second term in the geometric sequence are approximately:
[tex]\[
1.0953573965284054 \times 10^{-13} + 1788.854381999832j
\][/tex]
1. Identify the Known Terms:
We know the first term ([tex]\(a_1\)[/tex]) is 3125 and the third term ([tex]\(a_3\)[/tex]) is -1024.
2. Understand the Relationship in a Geometric Sequence:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio ([tex]\(r\)[/tex]). Given this relationship, the third term ([tex]\(a_3\)[/tex]) can be expressed as:
[tex]\[
a_3 = a_1 \cdot r^2
\][/tex]
3. Set Up the Equation:
Plug in the known values:
[tex]\[
-1024 = 3125 \cdot r^2
\][/tex]
4. Solve for the Common Ratio ([tex]\(r\)[/tex]):
We can solve for [tex]\(r^2\)[/tex]:
[tex]\[
r^2 = \frac{-1024}{3125}
\][/tex]
Simplifying it, we get:
[tex]\[
r^2 = -0.32768
\][/tex]
Solving for [tex]\(r\)[/tex], we take the square root of both sides:
[tex]\[
r = \sqrt{-0.32768}
\][/tex]
The result involves both real and imaginary parts, so we get:
[tex]\[
r \approx 3.5051436688908974 \times 10^{-17} + 0.5724334022399462j
\][/tex]
5. Find the Second Term ([tex]\(a_2\)[/tex]):
The second term ([tex]\(a_2\)[/tex]) is the first term multiplied by the common ratio [tex]\(r\)[/tex]:
[tex]\[
a_2 = a_1 \cdot r
\][/tex]
Substitute the values:
[tex]\[
a_2 = 3125 \cdot (3.5051436688908974 \times 10^{-17} + 0.5724334022399462j)
\][/tex]
Simplifying, we get:
[tex]\[
a_2 \approx 1.0953573965284054 \times 10^{-13} + 1788.854381999832j
\][/tex]
Therefore, the possible values for the missing second term in the geometric sequence are approximately:
[tex]\[
1.0953573965284054 \times 10^{-13} + 1788.854381999832j
\][/tex]
