Algebra Answers

We consider the sequence of real numbers (Un) defined on N by Uo = -1, U1 = 1/2 and for every n E N, U(n+2) = U(n+1) - 1/4 Un. Where N reprents the set of natural numbers. Vn = U(n+1) - (1/2)Un.
We define the sequence (Wn) by for every n E N, (Wn) = Un / Vn.
(i) Deduce an expression for Un in terms of n.
(ii) For every natural number n, we set Sn = Uo + U1 + U2 + ...+ Un. Show by induction that for every n E N, Sn = 2 - (2n + 3)/ 2^n.

A particle P, of mass m, is projected vertically upwards from a point O with speed 2g/k m/s in a medium whose resistance to motion is of magnitude kv per unit mass, where v is the speed of P and k a positive contant.
(i) Show that v(dv/dx) = -g - kv
The particle P attains the highest point H of its path.
(ii) Show further that OH = g/k^2 (2 - In3)metres
Given that P reaches another point M with speed g/2k m/s as it traces its path back to O, the point of projection,
(iii) Prove that HM = g/k^2 ( In2 - 1/2)

Three vectors a = (3i 4j), b = (-4i + 3j), c = (-3i + 4j) lie on a horizontal plane, pie(ll).
(i) Determine which of the vectors b or a is perpendicular to a.
Two smooth spheres, S and T, each of mass m lie on the plane ll. Sphere S is projected along the plane towards sphere T with velocity 5x(i + 2j) so that it collides obliquely with T. At the instant of collision, the line of impact is parallel to the vector a. Given that the coefficient of restitution between the two spheres is 1/2,
Show that
(ii) the component of the initial velocity of S parallel to a is 11x.
(iii) the component of the final velocity os S parallel to a is (11/4)x.
(iv) the magnitude of the impulse exerted by sphere T on sphere S is (33/4)mx.