Given that:
[tex]a_6=25\text{ and }a_8=6.25[/tex]
The general term of a geometric series with first term 'a' and common ratio 'r' is
[tex]a_n=ar^{n-1}[/tex]
So,
[tex]ar^5=25\text{ and }ar^7=6.25[/tex]
Divide them.
[tex]\begin{gathered} \frac{ar^7}{ar^5}=\frac{6.25}{25} \\ r^2=0.25 \\ r=\pm0.5 \end{gathered}[/tex]
Find a.
If r = 0.5,
[tex]\begin{gathered} a\cdot(0.5)^5=25 \\ a=800 \end{gathered}[/tex]
If r = -0.5,
[tex]\begin{gathered} a\cdot(-0.5)^5=25 \\ a=-800 \end{gathered}[/tex]
The first four terms of the geometric sequence with a = 800 and r = 0.5 are
[tex]\begin{gathered} a,ar,ar^2,ar^3=800,800\cdot(0.5),800\cdot(0.5)^2,800\cdot(0.5)^3 \\ =800,400,200,100 \end{gathered}[/tex]
The first four terms of the geometric sequence with a = -800 and r = -0.5 are
[tex]\begin{gathered} a,ar,ar^2,ar^3=-800,-800\cdot(-0.5),-800\cdot(-0.5)^2,-800\cdot(-0.5)^3 \\ =-800,400,-200,100 \end{gathered}[/tex]
