I'd like to explain a few things about how repentance and belief fit together. I do not believe that repentance (turning from sins) is a requirement for receiving eternal life. I have some reasons for this, I'd like to share a few.
What's even more interesting to me is that the Gospel of John does not mention that repentance is a requirement for receiving eternal life, only belief is. This is in spite of the fact that John's Gospel is THE book in the New Testament with the stated purpose of showing people how to receive eternal life. (John 20:30-31) In John, there are a number of people who suddenly believe without repenting first. (John 4:25-30, 8:30, 9:38) The most interesting of these is John 8:30 when it says:
Even as he spoke, many believed in him. (John 8:30)
"Believed in Him" is the phrase the Gospel of John uses to indicate that a person has experienced saving faith. So this means there were people who believed in Jesus who did not repent first.
The Apostle John, who wrote the Gospel that bears his name, used the word "repent" a number of times when he was writing to believers. So what he encourages people to do once they've believed is repent (turn away from) sins. (Revelation 2:5, 16, 21, 3:3, 19) The fact that he has to tell people to do it means that it doesn't happen automatically. That means that not everyone who believed in Jesus follows it with repentance. So here are a few scenarios that we can derive from all of this.
Unbelievers can repent and remain unbelievers.
Unbelievers can repent and then later become believers.
Unbelievers can become believers without repenting first.
Believers can refuse to repent and remain believers.
Believers can repent.
Repentance sometimes precedes belief, and sometimes it doesn't. There are certainly times when repentance facilitates an ease of belief. So there are many examples where repentance precedes belief, but there are many where it doesn't.