根號相除的四則運算

問題：1 除以 (√2) - 1 是多少？

1 ÷ [(√2) - 1]
= 1 / [(√2) - 1]
= [(√2) + 1] / [(√2) - 1][(√2) + 1]
= [(√2) + 1] / [(√2)² - 1²]
= (√2) + 1


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問題：求等比級數2 + (-1) + (1/2) + (-1/4) + .... 前7項的和

第一項 a = 2
公比 r = -1/2
項數 n = 7

該等比級數前 7 項的和
= a (1 - rⁿ) / (1 - r)
= 2 x [1 - (-1/2)⁷] / [1 - (-1/2)]
= 2 x [1 + (1/128)] / [1 + (1/2)]
= (129/64) / (3/2)
= (129/64) x (2/3)
= 43/32
= 1.34375

1/(√ 2-1) 有理化

=(√ 2+1)/(√ 2-1)(√ 2+1)

=√ 2+1



求等比級數2+(-1)+1/2+(-1/4)+....前7項的和
Sn=a1(1-r^n)/(1-r)

r=-1/2

S7=2*[1-(-1/2)^7]/(1-(-1/2))=43/32