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Further Mathematics

A Level

Prove by induction that the sum of the first n integers can be written as (1/2)(n)(n+1).

For n = 1, the sum is given by (1/2)(1)(1+1), which gives 1, the expected result. We now assume that the statement is true for some k. If we look at k+1, the sum is given by 1 + 2 + ... + k + (k+1). Since...

Answered by Jason S. • Further Mathematics tutor

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Integral of ln x

xln(x) - x

Answered by Oscar G. • Further Mathematics tutor

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y = artanh(x/sqrt(1+x^2)) , find dy/dx

Some of these examiners quite like asking students to find the derivative of an inverse trig or hyperbolic function to try and catch someone off guard. The best way to approach these is to first multiply ...

Answered by Eugene L. • Further Mathematics tutor

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Show that the sum from 1 to n of 1/(2n+1)(2n-1) is equal to n/(2n+1) by Induction

First we check that this is true for n=1: S₁ = 1/(1x3) which is equal to n/(2n+1) for n=1 therefore S_n= n/(2n+1) is true for n = 1. Next assume that it is true for n=k. S_k

Answered by Jamie F. • Further Mathematics tutor

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Two planes have eqns r.(3i – 4j + 2k) = 5 and r = λ (2i + j + 5k) + μ(i – j – 2k), where λ and μ are scalar parameters. Find the acute angle between the planes, giving your answer to the nearest degree.

Summary of solution: To find the angle between the planes, we must find the normal vector to each plane and then use the scalar product to find the angle between these two normal vectors....

Answered by Daniel C. • Further Mathematics tutor

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