If n is a positive integer, then (3+1)2n−(3−1)2n is

(3+1)2n=2nC0(3)2n+2nC1(3)2n−1+2nC2(3)2n−2+…+2nC2n(3)2n−2nand (3−1)2n=2nC0(3)2n(−1)0+2nC1(3)2n−1(−1)1+2nC2(3)2n−2(−1)2+…+2nC2n(3)2n−2n(−1)2n∴ (3+1)2n−(3−1)2n=2 2nC1(3)2n−1+2nC3(3)2n−3+2nC5(3)2n−5+…+2nC2n−1(3)2n−(2n−1)which is most certainly an irrational number because of odd powers of 3 in each of the terms.