Q.The third term of G.P. is $4$. The product of its first $5$ terms is
(A) $4^3$
(B) $4^4$
(C) $4^5$
(D) None of these

The key idea here is understanding the structure of a Geometric Progression (G.P.) and how its terms relate to each other.

Let the first term of the G.P. be $a$ and the common ratio be $r$.

The terms of the G.P. are $a, ar, ar^2, ar^3, ar^4, \dots$.

1. Given that the third term is $4$, we have $ar^2 = 4$.
2. The product of the first $5$ terms, $P_5$, is: $P_5 = a \cdot (ar) \cdot (ar^2) \cdot (ar^3) \cdot (ar^4)$
3. Combine the terms: …

A Geometric Progression (G.P.) is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. If the first term is $a$ and the common ratio is $r$, the terms are $a, ar, ar^2, ar^3, \dots$.

When we need to find the product of terms in a G.P., especially an odd number of terms, there's a very elegant property we can use. Consider a G.P. with $n$ terms. If $n$ is odd, there is a unique middle term.

Why this approach works:

Let the five terms of the G.P. be $a_1, a_2, a_3, a_4, a_5$.

In a G.P., there's a symmetrical relationship between terms equidistant from the beginning and the end.

Specifically, $a_1 \cdot a_5 = a_2 \cdot a_4 = a_3 \cdot a_3 = a_3^2$.

This is because if the terms are $a, ar, ar^2, ar^3, ar^4$:

$a_1 \cdot a_5 = a \cdot (ar^4) = a^2 r^4$

$a_2 \cdot a_4 = (ar) \cdot (ar^3) = a^2 r^4$

$a_3 \cdot a_3 = (ar^2) \cdot (ar^2) = a^2 r^4$

So, the product of the first five terms is $P = a_1 \cdot a_2 \cdot a_3 \cdot a_4 \cdot a_5$.

We can group these terms: $P = (a_1 \cdot a_5) \cdot (a_2 \cdot a_4) \cdot a_3$.

Using the property above, we substitute: $P = (a_3^2) \cdot (a_3^2) \cdot a_3 = a_3^{2+2+1} = a_3^5$.

This means the product of the first $5$ terms is simply the third term raised to the power of $5$. This generalises: for $n$ terms, the product is (middle term)$^n$.

Now, let's apply this to the given problem.

1. Represent the terms of the G.P.:

   Let the first term of the G.P. be $a$ and the common ratio be $r$.

   The first five terms are:

   $T_1 = a$

   $T_2 = ar$

   $T_3 = ar^2$

   $T_4 = ar^3$

   $T_5 = ar^4$

2. Use the given information:

   We are given that the third term of the G.P. is $4$.

   So, $T_3 = ar^2 = 4$.

3. Calculate the product of the first $5$ terms:

   The product $P$ of the first $5$ terms is:

   $P = T_1 \cdot T_2 \cdot T_3 \cdot T_4 \cdot T_5$

   $P = (a) \cdot (ar) \cdot (ar^2) \cdot (ar^3) \cdot (ar^4)$ …