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解答 - 几何数列

公比是：

r = 0.6363636363636364

r=0.6363636363636364

该系列的和是：

s = 360

s=360

此系列的通用形式是：

a_{n} = 220 \cdot {0.6363636363636364}^{n - 1}

a_n=220*0.6363636363636364^(n-1)

这个序列的第n项是：

220, 140, 89.0909090909091, 56.69421487603305, 36.07813673929376, 22.958814288641484, 14.61015454731731, 9.297371075565561, 5.916508866268993, 3.7650510967166317

220,140,89.0909090909091,56.69421487603305,36.07813673929376,22.958814288641484,14.61015454731731,9.297371075565561,5.916508866268993,3.7650510967166317

查看步骤

逐步解答

1. 找到公比

通过将序列中的任何项除以前一项来找到公比：

$a_{2} a_{1} = \frac{140}{220} = 0.6363636363636364$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.6363636363636364$

2. 求和

5 个额外步骤

$s_{n} = a * ((1 - r^{n}) / (1 - r))$

要找到系列的和，将第一项： $a = 220$ 、公比： $r = 0.6363636363636364$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 220 * ((1 - {0.6363636363636364}^{2}) / (1 - 0.6363636363636364))$

$s_{2} = 220 * ((1 - 0.4049586776859504) / (1 - 0.6363636363636364))$

$s_{2} = 220 * (0.5950413223140496 / (1 - 0.6363636363636364))$

$s_{2} = 220 * (0.5950413223140496 / 0.36363636363636365)$

$s_{2} = 220 \cdot 1.6363636363636365$

$s_{2} = 360$

3. 找到通用形式

$a_{n} = a \cdot r^{n - 1}$

要找到系列的通用形式，将第一项： $a = 220$ 和公比： $r = 0.6363636363636364$ 插入几何级数的公式：

$a_{n} = 220 \cdot {0.6363636363636364}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 220$

$a_{2} = a_{1} \cdot r^{n - 1} = 220 \cdot {0.6363636363636364}^{2 - 1} = 220 \cdot {0.6363636363636364}^{1} = 220 \cdot 0.6363636363636364 = 140$

$a_{3} = a_{1} \cdot r^{n - 1} = 220 \cdot {0.6363636363636364}^{3 - 1} = 220 \cdot {0.6363636363636364}^{2} = 220 \cdot 0.4049586776859504 = 89.0909090909091$

$a_{4} = a_{1} \cdot r^{n - 1} = 220 \cdot {0.6363636363636364}^{4 - 1} = 220 \cdot {0.6363636363636364}^{3} = 220 \cdot 0.25770097670924114 = 56.69421487603305$

$a_{5} = a_{1} \cdot r^{n - 1} = 220 \cdot {0.6363636363636364}^{5 - 1} = 220 \cdot {0.6363636363636364}^{4} = 220 \cdot 0.16399153063315347 = 36.07813673929376$

$a_{6} = a_{1} \cdot r^{n - 1} = 220 \cdot {0.6363636363636364}^{6 - 1} = 220 \cdot {0.6363636363636364}^{5} = 220 \cdot 0.1043582467665522 = 22.958814288641484$

$a_{7} = a_{1} \cdot r^{n - 1} = 220 \cdot {0.6363636363636364}^{7 - 1} = 220 \cdot {0.6363636363636364}^{6} = 220 \cdot 0.06640979339689686 = 14.61015454731731$

$a_{8} = a_{1} \cdot r^{n - 1} = 220 \cdot {0.6363636363636364}^{8 - 1} = 220 \cdot {0.6363636363636364}^{7} = 220 \cdot 0.04226077761620709 = 9.297371075565561$

$a_{9} = a_{1} \cdot r^{n - 1} = 220 \cdot {0.6363636363636364}^{9 - 1} = 220 \cdot {0.6363636363636364}^{8} = 220 \cdot 0.026893222119404512 = 5.916508866268993$

$a_{10} = a_{1} \cdot r^{n - 1} = 220 \cdot {0.6363636363636364}^{10 - 1} = 220 \cdot {0.6363636363636364}^{9} = 220 \cdot 0.017113868621439234 = 3.7650510967166317$

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