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解答 - 几何数列

公比是：

r = 0.34615384615384615

r=0.34615384615384615

该系列的和是：

s = 209

s=209

此系列的通用形式是：

a_{n} = 156 \cdot {0.34615384615384615}^{n - 1}

a_n=156*0.34615384615384615^(n-1)

这个序列的第n项是：

156, 54, 18.69230769230769, 6.470414201183432, 2.2397587619481105, 0.7753011099051152, 0.26837346112100147, 0.09289850577265434, 0.03215717507514958, 0.011131329833705624

156,54,18.69230769230769,6.470414201183432,2.2397587619481105,0.7753011099051152,0.26837346112100147,0.09289850577265434,0.03215717507514958,0.011131329833705624

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：

$a_{2} a_{1} = \frac{54}{156} = 0.34615384615384615$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.34615384615384615$

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： $a = 156$ 、公比： $r = 0.34615384615384615$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 156 * ((1 - {0.34615384615384615}^{2}) / (1 - 0.34615384615384615))$

$s_{2} = 156 * ((1 - 0.11982248520710058) / (1 - 0.34615384615384615))$

$s_{2} = 156 * (0.8801775147928994 / (1 - 0.34615384615384615))$

$s_{2} = 156 * (0.8801775147928994 / 0.6538461538461539)$

$s_{2} = 156 \cdot 1.346153846153846$

$s_{2} = 209.99999999999997$

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： $a = 156$ 和公比： $r = 0.34615384615384615$ 插入几何级数的公式：

$a_{n} = 156 \cdot {0.34615384615384615}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 156$

$a_{2} = a_{1} \cdot r^{n - 1} = 156 \cdot {0.34615384615384615}^{2 - 1} = 156 \cdot {0.34615384615384615}^{1} = 156 \cdot 0.34615384615384615 = 54$

$a_{3} = a_{1} \cdot r^{n - 1} = 156 \cdot {0.34615384615384615}^{3 - 1} = 156 \cdot {0.34615384615384615}^{2} = 156 \cdot 0.11982248520710058 = 18.69230769230769$

$a_{4} = a_{1} \cdot r^{n - 1} = 156 \cdot {0.34615384615384615}^{4 - 1} = 156 \cdot {0.34615384615384615}^{3} = 156 \cdot 0.0414770141101502 = 6.470414201183432$

$a_{5} = a_{1} \cdot r^{n - 1} = 156 \cdot {0.34615384615384615}^{5 - 1} = 156 \cdot {0.34615384615384615}^{4} = 156 \cdot 0.014357427961205838 = 2.2397587619481105$

$a_{6} = a_{1} \cdot r^{n - 1} = 156 \cdot {0.34615384615384615}^{6 - 1} = 156 \cdot {0.34615384615384615}^{5} = 156 \cdot 0.004969878909648175 = 0.7753011099051152$

$a_{7} = a_{1} \cdot r^{n - 1} = 156 \cdot {0.34615384615384615}^{7 - 1} = 156 \cdot {0.34615384615384615}^{6} = 156 \cdot 0.001720342699493599 = 0.26837346112100147$

$a_{8} = a_{1} \cdot r^{n - 1} = 156 \cdot {0.34615384615384615}^{8 - 1} = 156 \cdot {0.34615384615384615}^{7} = 156 \cdot 0.0005955032421323996 = 0.09289850577265434$

$a_{9} = a_{1} \cdot r^{n - 1} = 156 \cdot {0.34615384615384615}^{9 - 1} = 156 \cdot {0.34615384615384615}^{8} = 156 \cdot 0.00020613573766121524 = 0.03215717507514958$

$a_{10} = a_{1} \cdot r^{n - 1} = 156 \cdot {0.34615384615384615}^{10 - 1} = 156 \cdot {0.34615384615384615}^{9} = 156 \cdot 7.13546784211899 E - 05 = 0.011131329833705624$

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几何数列