Not sure if the given probability that [tex]X=0[/tex] is 0.06 or 0.6, or something else altogether. I'll just refer to it by the number [tex]p[/tex].
[tex]\begin{cases}P(X=0)=p\\P(X=i+1)=\frac15P(X=i)&\text{for }i\ge1\end{cases}[/tex]
As is, the probability that [tex]X=1[/tex] is indeterminate, so I think you intended to write
[tex]\begin{cases}P(X=0)=p\\P(X=i+1)=\frac15P(X=i)&\text{for }i\ge\boxed{0}\end{cases}[/tex]
Then by this definition,
[tex]i=0\implies P(X=1)=\dfrac15P(X=0)=\dfrac p5[/tex]
[tex]i=1\implies P(X=2)=\dfrac15P(X=1)=\dfrac p{5^2}[/tex]
and so on.
Then
[tex]\boxed{P(X\le2)=P(X=0)+P(X=1)+P(X=2)=p+\dfrac p5+\dfrac p{5^2}=\dfrac{31p}{25}}[/tex]
