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解答 - 几何数列

公比是：

r = 0.3018867924528302

r=0.3018867924528302

该系列的和是：

s = 69

s=69

此系列的通用形式是：

a_{n} = 53 \cdot {0.3018867924528302}^{n - 1}

a_n=53*0.3018867924528302^(n-1)

这个序列的第n项是：

53, 16, 4.830188679245283, 1.4581701673193306, 0.4402023146624394, 0.1328912648037553, 0.040118117676605364, 0.012111129864635582, 0.0036561901478145155, 0.0011037555163213632

53,16,4.830188679245283,1.4581701673193306,0.4402023146624394,0.1328912648037553,0.040118117676605364,0.012111129864635582,0.0036561901478145155,0.0011037555163213632

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：

$a_{2} a_{1} = \frac{16}{53} = 0.3018867924528302$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.3018867924528302$

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： $a = 53$ 、公比： $r = 0.3018867924528302$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 53 * ((1 - {0.3018867924528302}^{2}) / (1 - 0.3018867924528302))$

$s_{2} = 53 * ((1 - 0.09113563545745816) / (1 - 0.3018867924528302))$

$s_{2} = 53 * (0.9088643645425418 / (1 - 0.3018867924528302))$

$s_{2} = 53 * (0.9088643645425418 / 0.6981132075471699)$

$s_{2} = 53 \cdot 1.3018867924528301$

$s_{2} = 69$

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： $a = 53$ 和公比： $r = 0.3018867924528302$ 插入几何级数的公式：

$a_{n} = 53 \cdot {0.3018867924528302}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 53$

$a_{2} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.3018867924528302}^{2 - 1} = 53 \cdot {0.3018867924528302}^{1} = 53 \cdot 0.3018867924528302 = 16$

$a_{3} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.3018867924528302}^{3 - 1} = 53 \cdot {0.3018867924528302}^{2} = 53 \cdot 0.09113563545745816 = 4.830188679245283$

$a_{4} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.3018867924528302}^{4 - 1} = 53 \cdot {0.3018867924528302}^{3} = 53 \cdot 0.027512644666402464 = 1.4581701673193306$

$a_{5} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.3018867924528302}^{5 - 1} = 53 \cdot {0.3018867924528302}^{4} = 53 \cdot 0.008305704050234706 = 0.4402023146624394$

$a_{6} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.3018867924528302}^{6 - 1} = 53 \cdot {0.3018867924528302}^{5} = 53 \cdot 0.0025073823547878357 = 0.1328912648037553$

$a_{7} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.3018867924528302}^{7 - 1} = 53 \cdot {0.3018867924528302}^{6} = 53 \cdot 0.0007569456165397239 = 0.040118117676605364$

$a_{8} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.3018867924528302}^{8 - 1} = 53 \cdot {0.3018867924528302}^{7} = 53 \cdot 0.00022851188423840722 = 0.012111129864635582$

$a_{9} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.3018867924528302}^{9 - 1} = 53 \cdot {0.3018867924528302}^{8} = 53 \cdot 6.89847197700852 E - 05 = 0.0036561901478145155$

$a_{10} = a_{1} \cdot r^{n - 1} = 53 \cdot {0.3018867924528302}^{10 - 1} = 53 \cdot {0.3018867924528302}^{9} = 53 \cdot 2.082557577964836 E - 05 = 0.0011037555163213632$

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几何数列