To solve this problem, let us denote the first term of the geometric series as [tex]\( a \)[/tex] and the common ratio as [tex]\( r \)[/tex].
We are given that the third term of the geometric series is 3. In a geometric series, the third term can be expressed as:
[tex]\[ a \cdot r^2 = 3 \][/tex]
Now, let's determine the product of the first five terms. The first five terms are:
1. [tex]\( a \)[/tex]
2. [tex]\( ar \)[/tex]
3. [tex]\( ar^2 \)[/tex]
4. [tex]\( ar^3 \)[/tex]
5. [tex]\( ar^4 \)[/tex]
The product of these terms is:
[tex]\[ a \cdot (ar) \cdot (ar^2) \cdot (ar^3) \cdot (ar^4) \][/tex]
Let's simplify this expression step by step:
[tex]\[ (a \cdot ar \cdot ar^2 \cdot ar^3 \cdot ar^4) \][/tex]
[tex]\[ = a \cdot a \cdot r \cdot a \cdot r^2 \cdot a \cdot r^3 \cdot a \cdot r^4 \][/tex]
[tex]\[ = a^5 \cdot r^{1+2+3+4} \][/tex]
[tex]\[ = a^5 \cdot r^{10} \][/tex]
From the given information, we know that:
[tex]\[ ar^2 = 3 \][/tex]
To find [tex]\( a^5 \cdot r^{10} \)[/tex], let's express it in terms of the given third term, [tex]\( ar^2 \)[/tex]. We'll use [tex]\( ar^2 = 3 \)[/tex] as a basis for our calculation.
We need to express [tex]\( a^5 \cdot r^{10} \)[/tex] in terms of [tex]\( ar^2 \)[/tex]:
[tex]\[ a^5 \cdot r^{10} = (ar^2)^5 \][/tex]
[tex]\[ = 3^5 \][/tex]
Thus, we need to calculate [tex]\( 3^5 \)[/tex]:
[tex]\[ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 \][/tex]
[tex]\[ = 243 \][/tex]
So the product of the first five terms of the geometric series is [tex]\( 243 \)[/tex].
