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begin mathsize 22px style bold The bold space bold third bold space bold term bold space bold of bold space bold a bold space bold GP bold space bold is bold space bold 3 bold. bold space bold The bold space bold product bold space bold of bold space bold the bold space
bold first bold space bold five bold space bold terms bold. end style

  216
  226
  243
  Cannot be determined

We are given that the third term of a Geometric Progression (GP) is 3, and we need to find the product of the first five terms of the GP.

### Let’s denote the terms of the GP:
- Let the first term be \( a \).
- Let the common ratio be \( r \).

The general formula for the \( n \)-th term of a GP is:
\[
T_n = a \cdot r^{n-1}
\]

Given:
\[
T_3 = a \cdot r^2 = 3
\]

Now, the product of the first five terms of the GP is:
\[
P = T_1 \cdot T_2 \cdot T_3 \cdot T_4 \cdot T_5
\]

Substituting the terms:
\[
P = (a \cdot a \cdot r) \cdot (a \cdot r^2) \cdot (a \cdot r^3) \cdot (a \cdot r^4)
\]
This simplifies to:
\[
P = a^5 \cdot r^{10}
\]

Now, we already know:
\[
a \cdot r^2 = 3
\]
Therefore:
\[
a^5 \cdot r^{10} = (a \cdot r^2)^5 = 3^5 = 243
\]

Thus, the product of the first five terms is \( 243 \).

### The correct answer is:
\[
\boxed{243}
\]

Would you like further details on this, or do you have any questions?

Here are 5 related questions:
1. How do you find the common ratio of a GP if the first term and second term are known?
2. What is the general formula for the sum of the first \( n \) terms of a GP?
3. How does the common ratio affect the growth or decay of a GP?
4. Can you solve a GP problem where only the sum of certain terms is given?
5. How does an infinite geometric series differ from a finite geometric series?

**Tip:** In a GP, the ratio between consecutive terms remains constant, making the geometric sequence predictable when terms are known.

The third term of a GP is 3. The product of the first five terms. 216, 226, 243, Cannot be determined

Learn how to solve a geometric progression problem where the third term is 3 and the product of the first five terms needs to be determined. This detailed solution includes key formulas like T_n = a * r^(n-1) and shows how to calculate the product of terms. Suitable for students in Grades 9-11.

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