Bunuel wrote:
1. For each positive integer k, the quantity \(x_k\) is defined such that \(x_k = x_{k+1} * ( x_{k+2})^3\)
In addition, \(x_3 = 1\). In the table, select values for \(x_1\) and \(x_4\) that are jointly compatible with these conditions. Select only two values, one in each column.
\(x_1\) -- \(x_4\)
------------------------------- 0
------------------------------- 2
------------------------------- 5
------------------------------- 8
------------------------------- 16
------------------------------- 64
From the above it can be deduced, that x1 = x2 = (x4)^3. Hence, for values of x1, x4 only 2,8 satisfy the same. If the above values apply, I believe the expression x-k cannot be said always an integer. since x-5 = (1/2)^(1/3). But since the question mentions it a 'quantity', it seems that fits the question as well.
Please correct me if I am wrong
Regards,
Arpan