Binomial

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


b)
1 + (1+x) + (1+x)^2 + ... + (1+x)^24
= 1[(1+x)^25 - 1] / (1+x - 1)
= [(1+x)^25 - 1] / x
= (1 + (25C1)x + (25C2)x^2 + (25C3)x^3 + ... + (25C25)x^25 - 1) / x
= (25C1) + (25C2)x + (25C3)x^2 + ... + (25C25)x^24
The coefficient of x^2 = 25C3 = 2300


2010-04-26 20:55:23 補充：
Why sub. r = 25?

1 + (1+x) + (1+x)^2 + ... + (1+x)^24

即

1 + (1+x) + (1+x)^2 + ... + (1+x)^(25-1)

如不選我,請通知我,讓我刪除答案,多謝!!

=.=

2010-04-26 20:57:09 補充：
真的是左右為難~

rS - S
= r[ a + ar + ... + ar^(n-1) ] - [ a + ar + ... + ar^(n-1) ]
= ar + ar^2 + ... + ar^n - a - ar - ... - ar^(n-1)
= ar^n - a
= a(r^n - 1)

rS - S = a(r^n - 1)
(r - 1)S = a(r^n - 1)
S = a(r^n - 1) / (r - 1)
---------------------------------------------------------------
a = 1, r = 1 + x, n = 25
S = [ (1 + x)^25 - 1 ] / (1 + x - 1)
= [ (1 + x)^25 - 1 ] / x
= [ 1 + 25x + .. + (25C3) x^3 + ... - 1 ] / x
= [ 25x + .. + (25C3) x^3 + ... ] / x
= 25 + .. + (25C3) x^2 + ...

coeff of x^2 = 25C3 = 2300


2010-04-26 20:37:41 補充：
好似我清楚 d, 請投我一票, 謝謝