Alan

As follows:

Looking at the first few numbers in the sequence we see that the remainders cycle in the order 1, 2, 2, 4, 3, 2 with period 6. 

So we just need to find the remainder when 1000 is divided by 6. This is 4.  The numbers above correspond to remainders of 0, 1, 2, 3, 4, 5 respectively, so the remainder when a₁₀₀₀ is divided by 5 is 3.

"The cubic polynomial x^3 + ax^2 + b has three roots 2, 4, and r.  Find a and b."

"Let $f(x)$ be a polynomial such that $f(0) = 4,$ $f(1) = 5,$ and $f(2) = 10.$ Find the remainder when $f(x)$ is divided by $x(x - 1)(x - 2).$"

Managed to do this graphically using a package called Autograph:

Looks like the minimum is 48.

Maximum value found as follows:

Looks like 7 is a factor of all of them (leap of faith based on the prime factorisation of ten values of n  !!):

(Melody, the value with n = 2 should be 28567 not 28267).

As follows:

Oops!  You are quite right.  My mistake.  I should have given it more thought instead of jumping to a quick conclusion.

"If x + y = 1 and x^2 + y^2 = 4, what is x^3 + y^3 equal to?"