MATH

1. Let triangle ABC be a triangle such that angle ACB is 135 degrees.

Prove that AB^2 = AC^2 + BC^2 - (Root 2) x AC x BC

2. Simplify:

2/1! - 3/2! + 4/3! - 5/4! + ..... + 2010/2009! - 2011/2010!

Where k! is the multiplication of all numbers up to k (e.g. 5! = 5x4x3x2x1=120)

3. For the algebra nerds:

Factorise:

a^4 + b^4 + c^2 - 2 (a^2 x b^2 + a^2 x c + b^2 x c)

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asked by Gerald
  1. 1. Let triangle ABC be a triangle such that angle ACB is 135 degrees.

    Prove that AB^2 = AC^2 + BC^2 - (Root 2) x AC x BC


    There is probably an error in the sign, because the result is obtained directly by application of the cosine law, and substituting cos(135°)=-√2
    AB^2 = AC^2 + BC^2 - 2 AC x BC cos(∠ACB)
    =AC^2 + BC^2 + 2√2 AC x BC

    So there must have been a misprint in the original question.



    2. Simplify:

    2/1! - 3/2! + 4/3! - 5/4! + ..... + 2010/2009! - 2011/2010!

    Where k! is the multiplication of all numbers up to k (e.g. 5! = 5x4x3x2x1=120)


    With a little patience, it can be shown that the sum of the sequence having the general term k:
    (-1)^k * k/(k-1)!
    for k=2 to n is
    ((n-1)!+(-1)^k)/(k-1)!

    Thus for n=2011,
    the sum is (2010!+1)/2010!

    3. For the algebra nerds:

    Factorise:

    a^4 + b^4 + c^2 - 2 (a^2 x b^2 + a^2 x c + b^2 x c)


    The expression factorizes to
    Rearrange the expression to
    a^4+b^4+2a²b²+c²-2a²c-2b²c) - 4a²b²
    =(c-b²-a²)² -(2ab)²
    =(c-b^2-2*a*b-a^2)*(c-b^2+2*a*b-a^2)

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  2. Correction:
    For the general sum from k=2 to k=n

    Sum=((n-1)!+(-1)^n)/(n-1)!

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