For 2k real numbers a1,a2,...,ak, b1,b2,...,bk define a sequence of numbers Xn by
X_n = \sum_{i=1}^k [a_in + b_i] (n=1,2,...).
If the sequence XN forms an arithmetic progression, show that ∑i=1kai must be an integer. Here [r] denotes the greatest integer less than or equal to r. floor functionarithmetic sequencealgebra