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.