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The problem asks to:

**Find the 50th term of a GP (Geometric Progression) if the 20th term is 1200, and the 30th term is also 1200.**

### Let's break this down:

For a GP, the \( n \)-th term is given by:

\[
T_n = a \cdot r^{n-1}
\]

Where:
- \( T_n \) is the \( n \)-th term,
- \( a \) is the first term,
- \( r \) is the common ratio.

We are given:
- \( T_{20} = 1200 \)
- \( T_{30} = 1200 \)

### Step 1: Set up the equations for the 20th and 30th terms.
Using the formula for the \( n \)-th term:

\[
T_{20} = a \cdot r^{19} = 1200
\]
\[
T_{30} = a \cdot r^{29} = 1200
\]

### Step 2: Divide the two equations.
To eliminate \( a \), divide \( T_{30} \) by \( T_{20} \):

\[
\frac{a \cdot r^{29}}{a \cdot r^{19}} = \frac{1200}{1200}
\]
\[
r^{10} = 1
\]

### Step 3: Solve for \( r \).
Since \( r^{10} = 1 \), the possible values for \( r \) are the 10th roots of unity, which means \( r = 1 \) or \( r = -1 \) (real solutions).

### Step 4: Use \( r = 1 \) or \( r = -1 \).
#### Case 1: \( r = 1 \)
If \( r = 1 \), the terms of the GP are constant, meaning \( T_n = 1200 \) for all \( n \). Therefore, the 50th term is:

\[
T_{50} = 1200
\]

#### Case 2: \( r = -1 \)
If \( r = -1 \), the terms alternate between 1200 and -1200. In this case, we alternate signs for even and odd terms. Since \( T_{20} = 1200 \), the terms at even positions are 1200, and the terms at odd positions are -1200. Therefore, \( T_{50} \) will be:

\[
T_{50} = 1200
\]

### Conclusion:
In both cases, the 50th term is \( \boxed{1200} \).

Would you like further clarification or a detailed explanation of any specific part? 

Here are some related questions:
1. What is the general form of a geometric progression?
2. How do you derive the 10th roots of unity in general?
3. What are some real-world applications of geometric progressions?
4. Can you explain how alternating terms work in a GP with \( r = -1 \)?
5. How do the properties of GPs change with different values of \( r \)?

**Tip:** For geometric progressions, analyzing the ratio \( r \) is key to understanding the pattern of the sequence.

Find the 50th term of a GP if the 20th term is 1200, and the 30th term is also 1200.

Learn how to find the 50th term of a geometric progression where the 20th and 30th terms are both 1200. Explore key concepts such as the geometric sequence formula and roots of unity. This step-by-step solution is ideal for students in Grades 10-12.

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