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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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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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Prove that 27(23^n)+17(10^2n)+22n is divisible by 11 for n belongs to the natural numbers

Proofs come in many shapes and sizes but this one is done via induction. When doing proof by induction, examiners love a list of a clear definite list of steps, the first being the base case, in this case it...
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