Proof question: "Prove that if 𝑛 is an odd integer then ⌈ (2𝑛+1) / 2 ⌉ = 𝑛 + 1"

Question

Am I wrong in thinking that the 2's in the numerator and denominator should cancel each other out, leaving me with ⌈𝑛+1⌉= n + 1, and because n + 1 is an integer, that it's ceiling is simply n + 1? Or am I approaching it the wrong way?

Answer 1

The ceiling function maps a real number to the smallest integer that follows it.

 

Consider (2n + 1)/2, with n odd.

 

(2n + 1)/2 = 2n/2 + 1/2 = n + 1/2

 

The integer that follows (n + 1/2) is (n + 1).  It seems to me this is true regardless of whether n is even or odd, because n is an integer, (n + 1/2) is not in integer, so (n + 1) is the next integer after (n + 1/2).