the  first term of a geometric progression is 2.The common ratio of G.P is also 2.The production of the last two terms of the G.Pis 512.Determine the number of terms in the G.p

To solve for the number of terms in this geometric progression (G.P.), we need to break the problem down step by step.

### Given:
- The first term \( a_1 = 2 \).
- The common ratio \( r = 2 \).
- The product of the last two terms of the G.P. is 512.

### Formula for the nth term of a G.P.:
The nth term of a geometric progression can be given by:
\[
a_n = a_1 \cdot r^{n-1}
\]
Thus, in this case, the nth term is:
\[
a_n = 2 \cdot 2^{n-1} = 2^n
\]

Now, let’s define the last two terms:
- The last term, \( a_n = 2^n \),
- The second-to-last term, \( a_{n-1} = 2^{n-1} \).

### Product of the last two terms:
We are told the product of the last two terms is 512. So:
\[
a_n \cdot a_{n-1} = 512
\]
Substituting the expressions for \( a_n \) and \( a_{n-1} \):
\[
2^n \cdot 2^{n-1} = 512
\]
Using the property of exponents:
\[
2^{n + (n-1)} = 512
\]
\[
2^{2n-1} = 512
\]
Now express 512 as a power of 2:
\[
512 = 2^9
\]
So:
\[
2^{2n-1} = 2^9
\]
Equating the exponents:
\[
2n - 1 = 9
\]
Solve for \( n \):
\[
2n = 10
\]
\[
n = 5
\]

Thus, the number of terms in the geometric progression is **5**.

Would you like further details or clarifications?

### Related questions:
1. How do we generalize the formula for the nth term in a geometric progression?
2. What are the applications of geometric progressions in real-life scenarios?
3. How do you find the sum of the first \(n\) terms in a geometric progression?
4. Can the product of any two consecutive terms in a G.P. give additional insights?
5. What is the difference between arithmetic and geometric progressions?

### Tip:
In geometric progressions, the ratio between consecutive terms remains constant, which simplifies solving for missing terms using powers of the common ratio.

The first term of a geometric progression is 2. The common ratio of G.P is also 2. The product of the last two terms of the G.P is 512. Determine the number of terms in the G.P.

Learn how to find the number of terms in a geometric progression where the first term is 2, the common ratio is 2, and the product of the last two terms is 512. This problem covers key concepts like geometric progressions and exponent properties, suitable for students in Grades 9-12.

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