## [數學] 幫幫忙,求解釋

### [數學] 幫幫忙,求解釋

Perform a geometric expansion to simplify the following series:
1,200 = 100(1 + x + x2 + x3 + . . . + x9).
What is the value of x?

[ANS]
1200 = 100(1 + x + x2 + x3 + . . . + x9)
1200x = 100(x + x2 + x3 + . . . + x10)
1200x - 1200 = 100(x + x2 + x3 + . . . + x10 − 1 − x − x2 − x3 − . . . − x9)
1200(x - 1) = 100(x10 - 1)
12 = (x10 - 1)/(x - 1)
Substitute for x:
x = 1.0398905

jiasu

### Re: [數學] 幫幫忙,求解釋

jiasu 寫到:Perform a geometric expansion to simplify the following series:
1,200 = 100(1 + x + x2 + x3 + . . . + x9).
What is the value of x?

[ANS]
1200 = 100(1 + x + x2 + x3 + . . . + x9)
1200x = 100(x + x2 + x3 + . . . + x10)
1200x - 1200 = 100(x + x2 + x3 + . . . + x10 − 1 − x − x2 − x3 − . . . − x9)
1200(x - 1) = 100(x10 - 1)
12 = (x10 - 1)/(x - 1)
Substitute for x:
x = 1.0398905

1,200 = 100(1 + x + x2 + x3 + . . . + x9)

1200(x - 1) = 100(x10 - 1)

f(x) = -11 + x + x2 + x3 + . . . + x9
g(x) = x10 - 12x + 11

f(1) = -2 小於 0
f(2) >0

g(1+a) = (1+a)10 - 12(1+a) + 11 = (1+a)10 - 12a - 1
≈(1+ 10a + 45a2) - 12a - 1 = 45a2 - 2a = 0

(1+a)10≈ 1 + 10a + 45 a2 + 120 a3
g(1+ a) =0 ≈ 120 a3 + 45a2 - 2a
a ≈ 0.0401465

lskuo

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