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Prove, by induction, that 4^(n+1) + 5^(2n-1) is always divisible by 21

Firstly when proving something by induction, we always show that the base case works, i.e. we plug in n=1. In this case we get 4 2 + 5 1 =21, which is divisible by 21. Next we state what is called the induct...
GS
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A line has Cartesian equations x−p = (y+2)/q = 3−z and a plane has equation r ∙ [1,−1,−2] = −3. In the case where the angle θ between the line and the plane satisfies sin⁡θ=1/√6 and the line intersects the plane at z = 0. Find p and q.

This question gives hints as to how you should solve it. Because it tells you the angle between the line and the plane, this should indicate that doing something to do with finding the angle between the line...
JO
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Work out the equation of the tangent to the curve y=x^2+5x-8 at the point (2,6)

Firstly, we know that dy/dx is the gradient of the tangent to a curve. If we know the gradient of the tangent, and the point on the curve where the tangent touches it, we can work out the equation of the tan...
AD
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Find the general solution to the differential equation; y'' + 4y' = 24x^2

The first step in tackling any second order differential equation is to find the Auxiliary Function . Looking at the equation's left hand side, we can find the derivative terms - we will substitute the deriv...
RU
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Using de Moivre's theorem demonstrate that "sin6x+sin2x(16(sinx)^4-16(sinx)^2+3)"

Consider, (cosX+isinX) 6 using binomial expansion we find that this = "cos 6 X + 6icos 5 XsinX - 15cos 4 Xsin 2 X - 20icos 3 Xsin 3 X + 15cos 2 Xsin 4 X + 6icosXsin 5 X - sin 6 X". Next, by using d...
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