So,
Here we can identify the following terms of the given sequence:
[tex]\begin{gathered} a_3=50,000 \\ a_5=20,000,000 \end{gathered}[/tex]What we're going to do to find an explicit rule for this geometric sequence is to replace each term in the general form:
[tex]a_n=a_1(r)^{n-1}[/tex]We're given that:
[tex]\begin{gathered} a_3=50,000 \\ a_5=20,000,000 \end{gathered}[/tex]So, replacing we got:
[tex]\begin{gathered} 50,000=a_1(r)^{3-1}\to50,000=a_1(r)^2 \\ 20,000,000=a_1(r)^{5-1}_{}\to20,000,000=a_1(r)^4_{} \end{gathered}[/tex]As you can see, here we have the following system:
[tex]\begin{cases}50,000=a_1(r)^2 \\ 20,000,000=a_1(r)^4_{}\end{cases}[/tex]We could divide equation 2 by equation 1 to find the value of r:
[tex]\frac{20,000,000=a_1(r)^4_{}}{50,000=a_1(r)^2}\to400=(r)^2\to r=20[/tex]Now that we know that r=20, we could find the value of a1:
[tex]\begin{gathered} 50,000=a_1(20)^2 \\ 50,000=a_1(400) \\ a_1=\frac{50,000}{400}=125 \end{gathered}[/tex]Therefore, the explicit rule will be:
[tex]a_n=125(20)^{n-1}[/tex]