Why must it be the even term?

If the fifth term is $10$ , then the sixth term is $10r$ , the seventh term is $10r^2$ , the eighth term is $10r^3$ , and the ninth term is $10r^4$ ; and likewise the fourth term is $\frac{10}{r}$ , the third term is $\frac{10}{r^2}$ , the second term is $\frac{10}{r^3}$ , and the first term is $\frac{10}{r^4}$ .

The product of the first $9$ terms is then:

$\frac{10}{r^4} \cdot \frac{10}{r^3} \cdot \frac{10}{r^2} \cdot \frac{10}{r} \cdot 10 \cdot 10r \cdot 10r^2 \cdot 10r^3 \cdot 10r^4$

All the $r$ terms cancel out, leaving $10^9 = \boxed{1000000000}$ .

let the first term be $a$ , and the common ratio $r$ .

then we have $ar^4 =10$ .

the product of the first 9 terms is $ar\times ar^2\times .... \times ar^8 = a^9 r^{1+2+...+8}=a^9 r^{36} = \left(a r^4\right)^9 = \boxed{10^9}$

The $n$ th term of a geometric seqence is given by,

$n$ th term = $a{ r }^{ n-1 }$ (Where a is the first term, and r is the common ratio)

So it is given the 5th term is 10, hence we can rewrite it mathematically using the formula.

10= $a{ r }^{ 5-1 }=a{ r }^{ 4 }$ ......... (1)

Now, the first 9 terms of the sequence would be a, $a{ r }^{ 1 }$ , $a{ r }^{ 2 }$ , ...., $a{ r }^{ 8 }$ .

Thus, their product would be ${ a }^{ 9 }{ r }^{ 36 }$ which can be rewritten as, ${ (a{ r }^{ 4 }) }^{ 9 }$ .

Now in the above rearrangement, we can substitute the value from (1) and thus we obtain ${ 10 }^{ 9 }$ = $\boxed{1000000000}$

Let the first term and the common ratio of the geometric progression be $a$ and $r$ respectively. Then the fifth term is $ar^4 = 10$ . The product of the first nine terms of the geometric progression is given by:

$\begin{aligned} P & = a \times ar \times ar^2 \times ar^3 \times \cdots \times ar^8 \\ & = a^9 r^{1+2+3+\cdots + 8} = a^9 r^{\frac {8(8+1)}2} = a^9r^{36} = \left(ar^4\right)^9 = 10^9 = \boxed{1000000000} \end{aligned}$

First term of the progression is not given. Assuming this to be 1 and common ratio be r, we have to calculate r^36 given $r^9$ =10. The answer then is $10^4$ =10000