Exploring a Constant-Ratio Pattern in Binomial Differences for Powers of Two

In this note, I look at a curious pattern in the expression (a+b)^n - (a-b)^n. By expanding and factoring cases where the exponent n is a power of two (like 2, 4, 8, and 16), I found that the result always simplifies to a constant of 2n once the symmetric factors are removed. I also tested numbers that aren't powers of two—like 3, 5, and 6—where this pattern doesn't work at all. My goal is to share this observation and the math behind it in a way that is easy for other students to follow.