Simple harmonic motion of a spring?

1. The problem statement, all variables and given/known data
I had a lab today and one of my questions is:

"If you displace the end of the spring a little or a lot, in other words if you change the amplitude of the oscillation, the period will be the exact same. 1) How can this happen? 2) What is different about the motion of the spring? 3) How does this relate to the equations you used in this experiment?"

2. Relevant equations

3. The attempt at a solution
Here is my explanation, can you tell me if I am near correct?

k is constant, therefore if the displacement (x) is increased the force will also increase. This force relates to F=ma. m will also be constant, therefore for F to increase, acceleration must increase. It is stated in the question that the period (or time) will be the same, so if acceleration increases (a=dv/dt) and t does not change, v must increase.

1) It can happen because the laws of physics say so??
2) The velocity of the spring will increase.
3) The equations relate like described above.

What do you think?

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