To find the values of k for which the given system of equations has at least one solution, let's solve the system of equations:

1. x + y + z = 1
2. x + 2y + 4z = k
3. x + 4y + 10z = k^2

We can solve this system of equations using the method of Gaussian elimination.

Step 1: Multiply equation 1 by -1 and add it to equation 2.
- (x + y + z) + (x + 2y + 4z) = -1 + k
y + 3z = k - 1 ----(4)

Step 2: Multiply equation 1 by -1 and add it to equation 3.
- (x + y + z) + (x + 4y + 10z) = -1 + k^2
3y + 9z = k^2 - 1 ----(5)

Step 3: Multiply equation 4 by -3 and add it to equation 5.
-3(y + 3z) + (3y + 9z) = -3(k - 1) + (k^2 - 1)
-6z = -3k + 3 + k^2 - 1
-6z = k^2 - 3k + 2
6z = 3k - k^2 + 2 ----(6)

Step 4: Substitute equation 6 into equation 4.
y + 3(3k - k^2 + 2) = k - 1
y + 9k - 3k^2 + 6 = k - 1
-3k^2 + y + 9k + 7 = 0 ----(7)

Now, we have two equations:
-3k^2 + y + 9k + 7 = 0 ----(7)
6z = 3k - k^2 + 2 ----(6)

The system of equations (7) and (6) represents a quadratic equation in terms of k, with variables y and z. To have at least one solution, the discriminant of this quadratic equation must be equal to zero.

Step 5: Find the discriminant of equation (7).
Discriminant = (9k)^2 - 4(-3)(y + 7)
Discriminant = 81k^2 + 12(y + 7)

For the discriminant to be zero, we have:
81k^2 + 12(y + 7) = 0
81k^2 + 12y + 84 = 0 ----(8)

Step 6: Find the discriminant of equation (6).
Discriminant = (3k)^2 - 4(1)(3k - 2)
Discriminant = 9k^2 - 12k + 8

For the discriminant to be zero, we have:
9k^2 - 12k + 8 = 0 ----(9)

Now, we need to solve equations (8) and (9) simultaneously to find the values of k.

By solving equations (8) and (9), we get two values of k: k = 1