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Certainly. The nth student (Sn) changes the state of every nth locker - i.e. the multiples of n. If changed an odd number of times, the locker is open -- if even then closed.
i. how many closed afte...

Firstly, the polynomial is a cubic, and so we know what its graph looks like, hence it must have at least one real root, and a) is false. Computing a root would be time consuming, so instead we adopt a di...

(x^2 - 9)(x^2 + 1) < 0 solving the equation to get solutions to the equality (x^2 - 9)(x^2 + 1) = 0 : x = +/- 3 or x = +/- 1 now consider points either side of these x-intercepts... for x&...

We know that x_n > x_(n+1) is true if and only if x_n - x_(n+1) > 0 is true.So x_n - x_(n+1) = (n^3 − 9n^2 + 631) − ((n + 1)^3 − 9(n + 1)^2 + 631) = (n^3 − n^3 − 3n^2 − 3n − 1) − 9(n^2 − n^2 − 2n − ...

We can see that 104 = 2^3 * 13 = 2226, 30 = 2 + 2 + 26, and 108 = 22 + 226 + 2*26, so the coefficients agree with the Vieta's formulas, so the roots of the equation above are 2, 2, 26...