If the third term of a GP is 4, then the product of first five terms o

We are given that the third term of the GP is 4.
$T_3 = ar^2 = 4$.

We need to find the product of the first five terms of the GP.
Product P = $T_1 \times T_2 \times T_3 \times T_4 \times T_5$
P = $a \times (ar) \times (ar^2) \times (ar^3) \times (ar^4)$

Let’s group the ‘a’ terms and the ‘r’ terms:
P = $(a \times a \times a \times a \times a) \times (r^0 \times r^1 \times r^2 \times r^3 \times r^4)$
P = $a^5 \times r^{(0+1+2+3+4)}$
P = $a^5 \times r^{10}$

We can rewrite this expression by grouping $(ar^2)$ terms:
P = $a^5 \times r^{10} = (a^5 \times r^{10/5 \times 5}) = (a^5 \times r^{2 \times 5})$
P = $(a \times r^2)^5$
P = $(ar^2)^5$

We know that $ar^2 = 4$.
Substitute the value of $ar^2$ into the expression for P:
P = $(4)^5 = 4^5$.