الرَّحِيْمِ

A word in the Arabic language will either be 1) a verb or 2) a noun or 3) a particle – الرَّحِيْمِ is a noun

A noun may either be 1) Definite or indefinite 2) Masculine or feminine 3) Singular, dual or plural 4) Nominative, accusative or genitive

الرَّحِيْمُ is the natural state of this definite noun and it changes to الرَّحِيْمِ due to: 1) a preposition that precedes it or 2) a possessed noun that precedes it or 7) it’s a noun of dependent declension a) نَعْتٌ / صِفَطٌ adjectives b) التَّوْكِيْدُ emphatic c) المَعْطُوْفٌ conjunction d) البَدَلُ synonym

Extremely and continuously loving and merciful and dispenser of grace and love as a result of our deeds and supplications, generally pertaining to the life to come

Grammatical analysis of الرَّحِيْمِ
اسْمٌ مَجْرُوْرٌ بِالكَسْرَةِ الزَّاهِرَةِ
مَعْرِفَةٌمُفْرَدٌمُذَكَّرٌمَرْفُوْعٌر ح مبَاب سَمِعَفَعِيْلٌرَحِمَ يَرْحَمُاسْمٌالاِسْمُ الْمُبَالَغَةُ
DefiniteSingularMasculineNominativeRootEver MercifulPatternسNounIntensive noun
التَّوَابِعُاسْمٌ فَاعِلٌ
Dependent declensionActive participle
الصِّفَاتُ المُشَبَّهَةُ بِأَسْمَاءِ الْفَاعِلِ  
Adjective resembling participles 
Surah/Below are the 8 verses in the holy Quran containing the genitive, definite, intensive, active participleالرَّحِيْمِ
Verse
1-1In the name of Allah, Most Compassionate, Eternally Mercifulبِسْمِ اللهِ الرَّحْمَانِ الرَّحِيْمِ
1-3Most Compassionate, Eternally Mercifulالرَّحْمَانِ الرَّحِيْمِ
26-217
27-30
36-5
36-58
41-2A revelation from the Most Gracious, the Most Mercifulتَنْزِيْلٌ مِّنَ الرَّحْمَانِ الرَّحِيْمِ
41-32
114-0In the name of Allah, Most Compassionate, Eternally Mercifulبِسْمِ اللهِ الرَّحْمَانِ الرَّحِيْمِ