The recursive formula
[tex]a_{n+1} = 5a_n-3[/tex]
is telling you that the next term in the sequence is five times the previous one, minus 3.
So, if we start with [tex]a_1=4[/tex], we have to multiply it by 5 and subtract 3 to get the next:
[tex]a_2 = 5a_1-3 = 5\cdot 4 - 3 = 20-3 = 17[/tex]
And similarly we can obtain all the next terms:
[tex]a_3 = 5a_2-3 = 5\cdot 17 - 3 = 85-3 = 82[/tex]
[tex]a_4 = 5a_3-3 = 5\cdot 82 - 3 = 410-3 = 407[/tex]
[tex]a_5 = 5a_4-3 = 5\cdot 407 - 3 = 2035-3 = 2032[/tex]
