In a geometric progression sequence the second term is 4 and the forth term is 9, what is the third term

In a geometric progression, the terms are related by a constant ratio, denoted as \( r \). Let's denote the first term of the sequence as \( a \).

The terms of a geometric progression can be written as:
- First term: \( a \)
- Second term: \( a \cdot r \)
- Third term: \( a \cdot r^2 \)
- Fourth term: \( a \cdot r^3 \)

We are given that:
1. The second term is 4, so \( a \cdot r = 4 \)
2. The fourth term is 9, so \( a \cdot r^3 = 9 \)

We can now set up the following system of equations:

\[
a \cdot r = 4 \quad \text{(1)}
\]
\[
a \cdot r^3 = 9 \quad \text{(2)}
\]

To solve for \( r \), divide equation (2) by equation (1):

\[
\frac{a \cdot r^3}{a \cdot r} = \frac{9}{4}
\]
This simplifies to:
\[
r^2 = \frac{9}{4}
\]
Taking the square root of both sides:
\[
r = \pm \frac{3}{2}
\]

Now, substitute this value of \( r \) back into equation (1) to solve for \( a \):

If \( r = \frac{3}{2} \):
\[
a \cdot \frac{3}{2} = 4 \quad \Rightarrow \quad a = \frac{4 \cdot 2}{3} = \frac{8}{3}
\]

If \( r = -\frac{3}{2} \):
\[
a \cdot \left(-\frac{3}{2}\right) = 4 \quad \Rightarrow \quad a = \frac{4 \cdot 2}{-3} = \frac{-8}{3}
\]

Now, calculate the third term \( a \cdot r^2 \):

1. For \( r = \frac{3}{2} \):
\[
\text{Third term} = a \cdot r^2 = \frac{8}{3} \cdot \frac{9}{4} = 6
\]

2. For \( r = -\frac{3}{2} \):
\[
\text{Third term} = a \cdot r^2 = \frac{-8}{3} \cdot \frac{9}{4} = -6
\]

Thus, the third term is either \( 6 \) or \( -6 \), depending on the sign of the common ratio.

Would you like more details on how to interpret these results?

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Here are 5 related questions to explore:

1. How do you find the sum of a geometric progression?
2. What happens when the common ratio \( r \) is a negative value?
3. How do you determine the number of terms in a geometric progression?
4. How do you solve geometric progression problems when given non-consecutive terms?
5. Can a geometric progression have a common ratio greater than 1?

**Tip:** Always check if the common ratio is positive or negative in geometric progression problems, as it can affect the signs of the terms.

In a geometric progression sequence, the second term is 4 and the fourth term is 9, what is the third term?

Learn how to find the third term in a geometric progression where the second term is 4 and the fourth term is 9. This detailed solution uses algebra and properties of geometric progressions. Suitable for students in Grades 9-12.

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