in a geometric progression a4 * a5=1/64
a1*a2*a3*a4....*a13=?
A geometric progression is one of the form $\displaystyle a, ar, ar^2, ar^3,\ldots$. So, suppose $\displaystyle a_0 = a, a_1 = ar, a_2 = ar^2, a_3 = ar^3, a_4 = ar^4, a_5 = ar^5$. Then $\displaystyle a_4 a_5 = (ar^4)(ar^5) = a^2r^9 = \dfrac{1}{64}$. So, $\displaystyle \prod_{n=1}^{13} a_n = a^{13}r^{91} = (a^2r^9)^6(ar^{37}) = \left(\dfrac{1}{64}\right)^6(ar^{37})$. You do not have enough information to solve this.
Edit: We can actually do a little better. Since $\displaystyle a\neq 0$, we can solve for $\displaystyle r$. $\displaystyle a^2r^9 = \dfrac{1}{64} \Rightarrow r^9 = (8a)^{-2} \Rightarrow r = (8a)^{-2/9}$. Then $\displaystyle \prod_{n=1}^{13} a_n = a^{13}r^{91} = a^{13}(8a)^{-182/9} = a^{-65/9}2^{-182/3}$. That's the closest I can come to a solution.
Terms are integers. So, $\displaystyle a_3 = ar^3, a_5 = ar^5$. Suppose $\displaystyle a=1$. Then $\displaystyle a_5-a_3 = r^5-r^3=24$ has $\displaystyle r=2$ as a solution. For any integer $\displaystyle a>1$, there are no integer solutions to $\displaystyle a(r^5-r^3) = 24$. So, that must be the correct solution. Then $\displaystyle a_3+a_5 = r^3+r^5 = 2^3+2^5 = 8 + 32 = 40$.
Hello, parmis!
There are at least 4 possible solutions . . .
$\displaystyle \text{In a geometric progression: }\;a_4\cdot a_5\,=\,\tfrac{1}{64}$
$\displaystyle \text{Find: }\:P \;=\;a_1\cdot a_2\cdot a_3 \cdots a_{13}$
We are given: .$\displaystyle a_4\cdot a_5 \,=\,\tfrac{1}{64}$
I found four cases: .$\displaystyle (a_4,a_5) \:=\:\left(\tfrac{1}{16},\tfrac{1}{4}\right),\; \left(\tfrac{1}{4},\tfrac{1}{16}\right),\;\left( \text{-}\tfrac{1}{16},\text{-}\tfrac{1}{4}\right),\;\left(\text{-}\tfrac{1}{4},\text{-}\tfrac{1}{64}\right)$
Take the first case: .$\displaystyle \begin{Bmatrix}a_4 &=& \frac{1}{16} \\ a_5 &=& \frac{1}{4} \end{Bmatrix}$
We find that: .$\displaystyle a = \tfrac{1}{4^5},\;r = 4$
Then: .$\displaystyle P \;=\;a_1\cdot a_2\cdot a_3\cdot a_4 \cdots a_{13}$
. . . . . . . .$\displaystyle =\; (a)(ar)(ar^2)(ar^3) \cdots (ar^{12})$
. . . . . . . .$\displaystyle =\; a^{13}r^{78}$
. . . . . $\displaystyle P \;=\;\left(\tfrac{1}{4^5}\right)^{13}(4)^{78} \;=\;\tfrac{1}{4^{65}}\cdot4^{78}$
. . . . . . . $\displaystyle =\;4^{13} \;=\;67,\!108,\!864$
You can work out the other three solutions.