Etymology
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increase (n.)
late 14c., "action of increasing; results of an increasing," from increase (v.) or from verbs formed from the noun in Old French or Anglo-French. The stress shifted from 18c. to distinguish it from the verb.
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increase (v.)
mid-14c., encresen, "become greater in size or number" (intransitive); late 14c., "cause to grow, enlarge" (transitive), from Anglo-French encress-, Old French encreiss-, present participle stem of encreistre, from Latin increscere "to increase, to grow upon, grow over, swell, grow into," from in- "in" (from PIE root *en "in") + crescere "to grow" (from PIE root *ker- (2) "to grow"). Modern English restored the Latin spelling 16c. Related: Increased; increasing.
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encrease 
obsolete or archaic form of increase.
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increasingly (adv.)
late 14c., from increasing (see increase (v.)) + -ly (2).
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increment (n.)
mid-15c., "act or process of increasing," from Latin incrementum "growth, increase; an addition," from stem of increscere "to grow in or upon" (see increase (v.)). Meaning "amount of increase" first attested 1630s.
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*ker- (2)

Proto-Indo-European root meaning "to grow."

It forms all or part of: accretion; accrue; cereal; Ceres; concrete; create; creation; creature; Creole; crescendo; crescent; crew (n.) "group of soldiers;" croissant; cru; decrease; Dioscuri; excrescence; excrescent; griot; increase; Kore; procerity; procreate; procreation; recreate; recreation; recruit; sincere.

It is the hypothetical source of/evidence for its existence is provided by: Greek kouros "boy," korē "girl;" Latin crescere "come forth, spring up, grow, thrive, swell," Ceres, goddess of agriculture, creare "to bring forth, create, produce;" Armenian serem "bring forth," serim "be born."

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*en 
Proto-Indo-European root meaning "in."

It forms all or part of: and; atoll; dysentery; embargo; embarrass; embryo; empire; employ; en- (1) "in; into;" en- (2) "near, at, in, on, within;" enclave; endo-; enema; engine; enoptomancy; enter; enteric; enteritis; entero-; entice; ento-; entrails; envoy; envy; episode; esoteric; imbroglio; immolate; immure; impede; impend; impetus; important; impostor; impresario; impromptu; in; in- (2) "into, in, on, upon;" inchoate; incite; increase; inculcate; incumbent; industry; indigence; inflict; ingenuous; ingest; inly; inmost; inn; innate; inner; innuendo; inoculate; insignia; instant; intaglio; inter-; interim; interior; intern; internal; intestine; intimate (adj.) "closely acquainted, very familiar;" intra-; intricate; intrinsic; intro-; introduce; introduction; introit; introspect; invert; mesentery.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit antara- "interior;" Greek en "in," eis "into," endon "within;" Latin in "in, into," intro "inward," intra "inside, within;" Old Irish in, Welsh yn, Old Church Slavonic on-, Old English in "in, into," inne "within, inside."
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*aug- (1)

Proto-Indo-European root meaning "to increase." It forms all or part of: auction; augment; augmentative; augur; August; august; Augustus; author; authoritarian; authorize; auxiliary; auxin; eke (v.); inaugurate; nickname; waist; wax (v.1) "grow bigger or greater."

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit ojas- "strength," vaksayati "cause to grow;" Lithuanian augu, augti "to grow," aukštas "high, of superior rank;" Greek auxo "increase," auxein "to increase;" Gothic aukan "to grow, increase;" Latin augmentum "an increase, growth," augere "to increase, make big, enlarge, enrich;" Old English eacien "to increase," German wachsen, Gothic wahsjan "to grow, increase."

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additament (n.)
c. 1400, "anything added, an increase or increment," from Latin additamentum "an increase," from past participle stem of addere "to add" (see add).
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augment (v.)
Origin and meaning of augment
late 14c., "become more severe;" c. 1400, "to make larger; become larger," from Old French augmenter "increase, enhance" (14c.), from Late Latin augmentare "to increase," from Latin augmentum "an increase, growth," from augere "to increase, make big, enlarge, enrich," from PIE root *aug- (1) "to increase." Related: Augmented; augmenting. As a noun from early 15c.
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