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find the sum of r from 0 to n of : 1/((r+1)(r+2)(r+3))

The solution like almost every Methods of Differences questions first involves putting the fraction into partial sums.At this point you would get 3 fractions which can be tricky to deal with. Following what ...
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Using z=cos(θ)+isin(θ), find expressions for z^n-1/z^n and z^n+1/z^n

We make use of De Moivre's Theorem which states that (cos( θ)+isin(θ))^n=cos(nθ)+isin(nθ). z^n-1/z^n=cos(nθ)+ isin(nθ)-cos(-nθ)- isin(-nθ)=cos(nθ)+ isin(nθ)-cos(nθ)+ isin(nθ) (by trig relationships)=2isin(nθ...
BS
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Show that the points on an Argand diagram that represent the roots of ((z+1)/z)^6 = 1 lie on a straight line.

We want to simplify this equation to one that we know how to solve. If we let ((z+1)/z) = w, then we need to solve w^6 = 1, which is more familiar. Now we try to find the modulus and argument of w. w = re^(i...
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Find the complementary function to the second order differential equation d^2y/dx^2 - 5dy/dx + 6x = x^2

Use the auxiliary equation k 2 -5k+6=0. Solving this gives roots k=2 and k=3, which are real and distinct roots. This means that the complementary function is of the form y=Ae^(k 1 x)+Be^(k 2 x), where k 1 a...
SM
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Prove by induction that the sum from r=1 to n of (2r-1) is equal to n^2.

Firstly, we note that there are 4 key steps for any proof by induction. (1) Basis : check the statement works for n=1. (2) Assumption : assume the statement is true for n=k. (3) Induction : prove the stateme...
JL
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