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Prove by induction that (n^3)-n is divisible by 3 for all integers n>0 (typical fp1 problem)

Let P(n) be the statement “(n^3)-n is divisible by 3” First we’ll examine the base case: P(1) (n^3)-n=1^3-1=1-1=0 0=3*0, so is divisible by 3, and so P(1) is true Now assume for some k>0, an integer, that...
TN
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A football is kicked at 30 m/s at an angle of 20° to the horizontal. It travels towards the goal which is 25 m away. The crossbar of the goal is 2.44 m tall. (A) Does the ball go into the goal, hit the crossbar exactly, or go over the top?

With questions about projectiles, such as a ball travelling in the shape of a parabola through the air as in the above question, it is always good to split the problem into the horizontal and vertical compon...
ES
Answered by Ehsaan S. • Maths tutor
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Explore Shakespeare’s presentation of family in King Lear. You must relate your discussion to relevant contextual factors and ideas from your critical reading.

Shakespeare illustrates that at the beginning of the play, Lear’s views on how family love is expressed and measured is corrupt as his two insincere daughters are rewarded while his most loving and most love...
BS
Answered by Bethan S. • English tutor
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Consider f:R -> R, f = x/ sqrt(x^2+1). Prove that for any a between -1 and 1, f(x)=a has only one solution.

f'(x)=( sqrt (x^2+1) - x * ( x / sqrt (x^2 +1) ) ) / (x^2+1) = (x^2 + 1 + x^2) / ( (x^2 + 1) * sqrt ( x^2 + 1) ) = 1 / ( (x^2 + 1) * sqrt (x^2 + 1) ). f'(x) > 0 for any x => f is increasing. When x-&gt...
AC
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What is globalisation and how is it changing the role of the state?

Traditionally, the focus of state-hood was state-centric, states were expected to exercise full jurisdiction internally and depend on self-help; states were independent on the world stage. Similarly, the tra...
LH
Answered by Lujane H. • Politics tutor
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