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解答 - 几何数列

公比是：

r = 0.15384615384615385

r=0.15384615384615385

该系列的和是：

s = 44

s=44

此系列的通用形式是：

a_{n} = 39 \cdot {0.15384615384615385}^{n - 1}

a_n=39*0.15384615384615385^(n-1)

这个序列的第n项是：

39, 6, 0.9230769230769232, 0.14201183431952666, 0.021847974510696408, 0.003361226847799448, 0.0005171118227383766, 7.955566503667333 E - 05, 1.2239333082565128 E - 05, 1.882974320394635 E - 06

39,6,0.9230769230769232,0.14201183431952666,0.021847974510696408,0.003361226847799448,0.0005171118227383766,7.955566503667333E-05,1.2239333082565128E-05,1.882974320394635E-06

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：

$a_{2} a_{1} = \frac{6}{39} = 0.15384615384615385$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.15384615384615385$

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： $a = 39$ 、公比： $r = 0.15384615384615385$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 39 * ((1 - {0.15384615384615385}^{2}) / (1 - 0.15384615384615385))$

$s_{2} = 39 * ((1 - 0.02366863905325444) / (1 - 0.15384615384615385))$

$s_{2} = 39 * (0.9763313609467456 / (1 - 0.15384615384615385))$

$s_{2} = 39 * (0.9763313609467456 / 0.8461538461538461)$

$s_{2} = 39 \cdot 1.1538461538461537$

$s_{2} = 44.99999999999999$

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： $a = 39$ 和公比： $r = 0.15384615384615385$ 插入几何级数的公式：

$a_{n} = 39 \cdot {0.15384615384615385}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 39$

$a_{2} = a_{1} \cdot r^{n - 1} = 39 \cdot {0.15384615384615385}^{2 - 1} = 39 \cdot {0.15384615384615385}^{1} = 39 \cdot 0.15384615384615385 = 6$

$a_{3} = a_{1} \cdot r^{n - 1} = 39 \cdot {0.15384615384615385}^{3 - 1} = 39 \cdot {0.15384615384615385}^{2} = 39 \cdot 0.02366863905325444 = 0.9230769230769232$

$a_{4} = a_{1} \cdot r^{n - 1} = 39 \cdot {0.15384615384615385}^{4 - 1} = 39 \cdot {0.15384615384615385}^{3} = 39 \cdot 0.003641329085116068 = 0.14201183431952666$

$a_{5} = a_{1} \cdot r^{n - 1} = 39 \cdot {0.15384615384615385}^{5 - 1} = 39 \cdot {0.15384615384615385}^{4} = 39 \cdot 0.0005602044746332413 = 0.021847974510696408$

$a_{6} = a_{1} \cdot r^{n - 1} = 39 \cdot {0.15384615384615385}^{6 - 1} = 39 \cdot {0.15384615384615385}^{5} = 39 \cdot 8.618530378972943 E - 05 = 0.003361226847799448$

$a_{7} = a_{1} \cdot r^{n - 1} = 39 \cdot {0.15384615384615385}^{7 - 1} = 39 \cdot {0.15384615384615385}^{6} = 39 \cdot 1.325927750611222 E - 05 = 0.0005171118227383766$

$a_{8} = a_{1} \cdot r^{n - 1} = 39 \cdot {0.15384615384615385}^{8 - 1} = 39 \cdot {0.15384615384615385}^{7} = 39 \cdot 2.039888847094188 E - 06 = 7.955566503667333 E - 05$

$a_{9} = a_{1} \cdot r^{n - 1} = 39 \cdot {0.15384615384615385}^{9 - 1} = 39 \cdot {0.15384615384615385}^{8} = 39 \cdot 3.1382905339910584 E - 07 = 1.2239333082565128 E - 05$

$a_{10} = a_{1} \cdot r^{n - 1} = 39 \cdot {0.15384615384615385}^{10 - 1} = 39 \cdot {0.15384615384615385}^{9} = 39 \cdot 4.828139283063167 E - 08 = 1.882974320394635 E - 06$

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几何序列常用于解释数学、物理、工程、生物、经济、计算机科学、金融等领域的概念，因此它们是我们工具箱中非常有用的工具。例如，几何序列最常见的应用之一就是计算已经获得或未付的复利，这是与金融相关的最常见活动之一，可能意味着赚取或失去大量的金钱！其他应用包括但不仅限于计算概率、测算随时间变化的放射性以及设计建筑物。

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几何数列