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Prove that 8 times any triangle number is always 1 less than a square number

A triangle number is a number such that it is the sum of n consecutive integers, starting from 0. Eg 1, 1+2, 1+2+3... are the first 3 triangle numbers. The formula for the nth triangle number is well-known a...
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Co-ordinate Geometry A-level: The equation of a circle is x^2+y^2+6x-2y-10=0, find the centre and radius of the circle, the co-ordinates of point(s) where y=2x-3 meets the circle and hence state what we can deduce about the relationship between them.

We have that x 2 + y 2 + 6x - 2y - 10 = (x+3) 2 + (y-1) 2 - 20 = 0 (step was motivated by the equation of a general circle (x-a) 2 + (y-b) 2 = r 2 )We know that the general circle has centre at (a,b) and the...
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What are some criticisms of Augustine's Soul-Deciding Theodicy?

Peter Vardy claims that there is no way to justify dysteleology or excess suffering. Evil seems to be more definite than a mere ‘privation of good’ for example Child cruelty is not just the lack of Child sup...
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What are the differences between the psychological approaches and how to compare them?

The approaches differ in terms of views, how they were developed and how they are applied in modern psychology whether it be studies or treating patients. Since the actual content is typically easier to gras...
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To what extent was the 'Abbasid Revolution of 747 a Shi'ite movement?

The 'Abbasid revolution can be most strongly argued to have been a proto-Shi'a revolution wherein notions of the primacy of the family of prophet were emerging against the backdrop of civil unrest and disenc...
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