等比数列求和

_{^{如果公比为2，有k个元素：}}Sk = 2^{0+₂1+₂2+₂3+...+₂k-1}

2(Sk) = 2(2^{0+₂1+₂2+₂3+...+₂k-1)}

= ₂1+₂2+₂3+...+₂k-1+2^k

得：

Sk = 2(S1) - S1 = 2^k-2⁰ = 2^k-1

通用推导：

设公比为q

Sn = a1+a2+a3+...+an

= a1 + a1q + a1q²+a1q³+...+a1q^n-1

qSn = a1q + a1q²+a1q³+...+a1q^n-1+_a1qn

qSn- Sn = a1qⁿ-a1

Sn = a1(qⁿ-1)/(q-1)

再回头让q=2，a1=1，带入翻过来验证：

Sn = 2ⁿ-1

时间： 2024-10-25 16:06:03

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