they just added for the sake of using the sum of the geometric series. it is just manipulation, a trick we can use to our advantage. don't let this jar you too much, the proof is valid. in fact, a similar trick was used earlier in the proof, obviously you didn't have a problem with it then. when proving P(2), they claimed: . where did that 1 come from? they just added it. does that make sense? of course, surely if we are less than 3 we would be less than 3 + 1 = 4. it is the same trick used here. will be a positive number. so surely, if we are less than then we will be less than