Champagne Breakfast
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#1
Report Thread starter 13 years ago
#1
Could someone outline the proofs for:

Quadratic equation
Arithmetic Progressions (Series)
Geometric Progressions (Series)

Or alternatively link to sites which explain them. Thanks.
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Retropmot
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#2
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Arithmetic

(a) + (a + d) + (a + 2d) + ... + (a + (n-2)d) + (a + (n-1)d)
writen backward this is
(a + (n-1)d) + (a + (n-2)d) + ...+ (a + 2d)+ (a + d) + (a)

add these 2 together term by teram and you get
(2a + (n-1)d) + (2a + (n-1)d) + .... + (2a + (n-1)d)
n(2a + (n-1)d)
this is twice the total so the total is
(n/2)(2a + (n-1)d)
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Retropmot
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#3
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Geometric

S = a + ar + ar^2 + ... + ar^(n-1)

muliply this by r and you get

rS = ar + ar^2 + ar^2 + ... + ar^(n)

subtract the first from the second and you get
S - rS = a - ar^n = a(1 - r^n) (as all the middle terms are present in both series they cancel)

S(1-r) = a(1 - r^n)

so S = a(1 - r^n)/(1-r)
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Retropmot
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#4
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Quadratic expression
the quadratic expression proog is just completing the square on ax^2 + bx + c = 0

x^2 + b/ax + c/a = 0

(x + b/2a)^2 = -c/a + (b/2a)^2

(2ax + b)^2= b^2 - 4ac (just muliplying through by 4a^2)

2ax + b = +/-rt(b^2 - 4ac)

2ax = - b +/-rt(b^2 - 4ac)

x = [- b +/-rt(b^2 - 4ac)]/2a
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Champagne Breakfast
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#5
Report Thread starter 13 years ago
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Thanks
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