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解答 - 几何数列

公比是：

r = 0.37373737373737376

r=0.37373737373737376

该系列的和是：

s = 136

s=136

此系列的通用形式是：

a_{n} = 99 \cdot {0.37373737373737376}^{n - 1}

a_n=99*0.37373737373737376^(n-1)

这个序列的第n项是：

99, 37, 13.82828282828283, 5.168146107540048, 1.9315293533230482, 0.7218847078076039, 0.26979529483718534, 0.10083258493915008, 0.037684905482308614, 0.014084257604499181

99,37,13.82828282828283,5.168146107540048,1.9315293533230482,0.7218847078076039,0.26979529483718534,0.10083258493915008,0.037684905482308614,0.014084257604499181

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：

$a_{2} a_{1} = \frac{37}{99} = 0.37373737373737376$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.37373737373737376$

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： $a = 99$ 、公比： $r = 0.37373737373737376$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 99 * ((1 - {0.37373737373737376}^{2}) / (1 - 0.37373737373737376))$

$s_{2} = 99 * ((1 - 0.1396796245281094) / (1 - 0.37373737373737376))$

$s_{2} = 99 * (0.8603203754718907 / (1 - 0.37373737373737376))$

$s_{2} = 99 * (0.8603203754718907 / 0.6262626262626263)$

$s_{2} = 99 \cdot 1.3737373737373737$

$s_{2} = 136$

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： $a = 99$ 和公比： $r = 0.37373737373737376$ 插入几何级数的公式：

$a_{n} = 99 \cdot {0.37373737373737376}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 99$

$a_{2} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.37373737373737376}^{2 - 1} = 99 \cdot {0.37373737373737376}^{1} = 99 \cdot 0.37373737373737376 = 37$

$a_{3} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.37373737373737376}^{3 - 1} = 99 \cdot {0.37373737373737376}^{2} = 99 \cdot 0.1396796245281094 = 13.82828282828283$

$a_{4} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.37373737373737376}^{4 - 1} = 99 \cdot {0.37373737373737376}^{3} = 99 \cdot 0.05220349603575806 = 5.168146107540048$

$a_{5} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.37373737373737376}^{5 - 1} = 99 \cdot {0.37373737373737376}^{4} = 99 \cdot 0.01951039750831362 = 1.9315293533230482$

$a_{6} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.37373737373737376}^{6 - 1} = 99 \cdot {0.37373737373737376}^{5} = 99 \cdot 0.007291764725329332 = 0.7218847078076039$

$a_{7} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.37373737373737376}^{7 - 1} = 99 \cdot {0.37373737373737376}^{6} = 99 \cdot 0.002725204998355407 = 0.26979529483718534$

$a_{8} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.37373737373737376}^{8 - 1} = 99 \cdot {0.37373737373737376}^{7} = 99 \cdot 0.0010185109589813139 = 0.10083258493915008$

$a_{9} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.37373737373737376}^{9 - 1} = 99 \cdot {0.37373737373737376}^{8} = 99 \cdot 0.00038065561093241027 = 0.037684905482308614$

$a_{10} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.37373737373737376}^{10 - 1} = 99 \cdot {0.37373737373737376}^{9} = 99 \cdot 0.00014226522832827456 = 0.014084257604499181$

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几何数列