Hensel Lifting Explanation 
I'm trying to understand this example for my exam tomorrow, I understand everything apart from where the recurrence relation comes from, is there a set formula for this? Any help is greatly appreciated.

Let's do this slowly. At each stage we have a number $a_n$ with both $a_n\equiv 3\pmod 5$ and with $a_n^3\equiv2\pmod{5^n}$. We try $a_{n+1}=a_n+5^nt$. Then
$$a_{n+1}^3\equiv a_n^3+3\times 5^na_n^24
=2+(a_n^3-2)+3\times 5^na_n^2t\pmod{5^{n+1}}.$$
Now
$$3a_n^2\equiv3\times 3^2\equiv 2\pmod5$$
so
$$3\times 5^na_n^2\equiv2\times 5^n\pmod{5^{n+1}}.$$
Therefore
$$a_{n+1}^3-2\equiv(a_n^3-2)+2\times 5^nt\pmod{5^{n+1}}.$$
We need $a_{n+1}^3-2\equiv0\pmod{5^{n+1}}$. So we want
$$(a_n^3-2)+2\times 5^nt\equiv0\pmod{5^{n+1}}.$$
Multiplying by $2$, this is equivalent to
$$2(a_n^3-2)-5^nt\equiv0\pmod{5^{n+1}},$$
that is
$$5^nt\equiv 2(a_n^3-2)\pmod{5^{n+1}}.$$
So
$$a_{n+1}=a_n+5^nt\equiv a_n+2(a_n^3-2)\pmod{5^{n+1}}.$$
This is the recurrence formula!
