I am guessing as to why John focused solely on the idea of believing in his Gospel, and did not use the word "faith"; John's message presented the words and actions of Jesus Christ as evidence of Who Jesus is. We know the definition of "faith" from the Book of Hebrews which says, "Now faith is the substance of things hoped for, the evidence of things not seen" (Heb. 11:1). John did not need faith to believe in Jesus; he knew Him! And he wasn't asking people to have faith in Jesus; he was asking people to believe his testimony. His Gospel ended with these words: "And there are also many other things which Jesus did, the which, if they should be written every one, I suppose that even the world itself could not contain the books that should be written. Amen." John provided his readers with all that was necessary for them to believe in Jesus Christ! The result of believing is faith! This is also obvious in the last chapter of 1 John; he wrote: "These things have I written unto you that believe on the name of the Son of God; that ye may know that ye have eternal life...(5:13)!
Here are a few verses which contain both "faith" and "believe":
* Romans 3:22 - "Even the righteousness of God which is by faith of Jesus Christ unto all and upon all them that believe: for there is no difference."
* Romans 4:5 - "But to him that worketh not, but believeth on Him that justifieth the ungodly, his faith is counted for righteousness."
* 2 Corinthians 4:13 - "We having the same Spirit of faith, according as it is written, I believed, and therefore have I spoken; we also believe, and therefore speak."
* Galatians 2:16 - "Knowing that a man is not justified by the works of the law, but by the faith of Jesus Christ, even we have believed in Jesus Christ, that we might be justified by the faith of Christ, and not by the works of the law: for by the works of the law shall no flesh be justified."
* Galatians 3:22 - "But the Scripture hath concluded all under sin, that the promise by faith of Jesus Christ might be given to them that believe."
* Hebrews 11:6 - "But without faith it is impossible to please Him: for He that cometh to God must believe that He is, and that He is a rewarder of them that diligently seek Him."
* 1 Peter 1:21 - "Who by Him do believe in God, that raised Him up from the dead, and gave Him glory; that your faith and hope might be in God."
Believe Jesus died for your sins, that He was buried,
and have faith in the One Who raised Him up the third day!