Explanations:

The letter's closing follows a typical pattern including a summary purpose and exhortation, personal greetings and a benediction.

v.12, I have written... exhorting & declaring... stand firm in it.  While the second clause could easily be defined as the purpose of the first (i.e. I have written in order to exhort and declare), I see this clause describing the style or form, that is the Manner of Peter's writing. His aim (Purpose) is expressed by the exhortation, stand firm.  He has written of true grace that calls for active faithfulness.

Cross References:

v.13, "She who is at Babylon, who is likewise chosen..." see 1:1, "to those who are elect exiles of the dispersion."

If Babylon is a reference to Rome, then this greeting is particularly meaningful to Peter's original readers who are out in the fringes of the Christian world. Much of this letter aims at encouraging his readers that they do indeed belong to the people of God (1:2-5; 2:9-10) and are not alone in the trials they are experiencing (5:9), so a personal greeting from the church in Rome is encouragement that they are not forgotten.

A Personal Note:

Arcing has been for me one of the single most effective exegetical tools I have ever used. It has sharpened my study of Scripture, increased my joy of discovery and improved my teaching. Arcing through 1st Peter for this blog has been a great joy and has itself been a wonderful means of honing my arcing skills. I am grateful for those who work hard to give us the online tool 'BibleArc' - what a great way to create, archive and share our arcs. When I began arcing a few years ago, what I longed for most was to interact with others, to learn from their insights, to ask questions of more experienced arcers, to challenge and be challenged. BibleArc has given us a great means to experience all of this and much more. Thank you Andy, Sarah and any unnamed laborers in the BibleArc 'cloud.'  I thank God for you.

Peace to all of you who are in Christ,  Mike