The word "incentive" comes from the Late Latin word incentivus, meaning "setting the tune", "inciting", or "provocative".   This is derived from incinere, meaning "to sing" or "to strike up a tune," which combines the prefix in- and the verb canere, "to sing". The meaning was influenced by the Latin word incendere, meaning "to set on fire" or "to kindle". 

According to the Oxford English Dictionary (OED),  the word "incentive" was first used in English before 1475, during the Middle English period.  The earliest evidence of its use is found in R. Higden's Polychronicon.  In reference to a system of rewards meant to encourage harder work, first attested 1943 in jargon of the U.S. war economy.  The earliest known use of the verb incentivize is in the 1960s. OED's earliest evidence for incentivize is from 1968, in the Guardian. incentivize is formed within English, by derivation. Etymons: incentive n., ‑ize suffix.

Examples of "incentive" used in a sentence: 
"The rising cost of electricity provides a strong incentive to conserve energy." 
"The company offered a bonus as an incentive to employees who exceed their sales goals." 
"As an incentive to motivate their children to do better in school, the parents offered a system of monetary incentives for grades received on report cards." 
"The government offers special tax incentives for entrepreneurs."   
"She offered to buy him dinner as an incentive to help her with the project." 
"There is little incentive for the government to invest in that area."