Haunting Problems

1. For integer n
f(n) = 1^n + 2^(n-1) + 3^(n-2) + … + (n-3)^3 + (n-1)^2 + n
What is the minimum value of f(n+1)/f(n), n>=6 ?

2. Given p(x) = x^2 + ax + b , a , b and n are integer. Find M that correspond with p(n) p(n+1) = p(M) .

Please!!!!!!!!!!!!!!!!!!!!

I think you made a mistake on the first question, shouldn't it be (n-3)^2 ?

more like (n-2)^3

For the second one. If you put work out p(n)p(n+1) and collect together coefficients you get:

n^4 + (2a+2)n³ + (a²+3a+2b+1)n² + (a²+2ab+a+2b)n + (b²+b+ab)

Now we want the function M such that p(M) is equal to this.
Straight away the must must have a highest power of n² as there is an n^4 term in what we are aiming for. So if we try say p(n²+dn+q) and collect coefficients we get:

n^4 + (2d)n³ + (a+2q+d²)n² + (2dq + ad)n + (q²+aq+b)

Comparing coefficients:

2a+2 = 2d
so d = a+1

a²+3a+2b+1 = a+2q+d² = a²+3a+2q+1
so 2b+1 = 2q+1
and q=b

You can check the other coefficients (I won't do it here but they do work out correctly).

So the function M = n² + (a+1)n + b

Just out of interest. Where are these problems from?