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解答 - 几何数列

公比是：

r = 0.3939393939393939

r=0.3939393939393939

该系列的和是：

s = 45

s=45

此系列的通用形式是：

a_{n} = 33 \cdot {0.3939393939393939}^{n - 1}

a_n=33*0.3939393939393939^(n-1)

这个序列的第n项是：

33, 13, 5.121212121212121, 2.017447199265381, 0.7947519269833319, 0.3130840924479792, 0.1233361576310221, 0.048586971187978396, 0.019140321983143003, 0.0075401268418442136

33,13,5.121212121212121,2.017447199265381,0.7947519269833319,0.3130840924479792,0.1233361576310221,0.048586971187978396,0.019140321983143003,0.0075401268418442136

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：

$a_{2} a_{1} = \frac{13}{33} = 0.3939393939393939$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.3939393939393939$

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： $a = 33$ 、公比： $r = 0.3939393939393939$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 33 * ((1 - {0.3939393939393939}^{2}) / (1 - 0.3939393939393939))$

$s_{2} = 33 * ((1 - 0.155188246097337) / (1 - 0.3939393939393939))$

$s_{2} = 33 * (0.8448117539026629 / (1 - 0.3939393939393939))$

$s_{2} = 33 * (0.8448117539026629 / 0.6060606060606061)$

$s_{2} = 33 \cdot 1.3939393939393938$

$s_{2} = 45.99999999999999$

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： $a = 33$ 和公比： $r = 0.3939393939393939$ 插入几何级数的公式：

$a_{n} = 33 \cdot {0.3939393939393939}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 33$

$a_{2} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.3939393939393939}^{2 - 1} = 33 \cdot {0.3939393939393939}^{1} = 33 \cdot 0.3939393939393939 = 13$

$a_{3} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.3939393939393939}^{3 - 1} = 33 \cdot {0.3939393939393939}^{2} = 33 \cdot 0.155188246097337 = 5.121212121212121$

$a_{4} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.3939393939393939}^{4 - 1} = 33 \cdot {0.3939393939393939}^{3} = 33 \cdot 0.06113476361410245 = 2.017447199265381$

$a_{5} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.3939393939393939}^{5 - 1} = 33 \cdot {0.3939393939393939}^{4} = 33 \cdot 0.02408339172676763 = 0.7947519269833319$

$a_{6} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.3939393939393939}^{6 - 1} = 33 \cdot {0.3939393939393939}^{5} = 33 \cdot 0.009487396740847854 = 0.3130840924479792$

$a_{7} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.3939393939393939}^{7 - 1} = 33 \cdot {0.3939393939393939}^{6} = 33 \cdot 0.003737459322152185 = 0.1233361576310221$

$a_{8} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.3939393939393939}^{8 - 1} = 33 \cdot {0.3939393939393939}^{7} = 33 \cdot 0.0014723324602417696 = 0.048586971187978396$

$a_{9} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.3939393939393939}^{9 - 1} = 33 \cdot {0.3939393939393939}^{8} = 33 \cdot 0.0005800097570649395 = 0.019140321983143003$

$a_{10} = a_{1} \cdot r^{n - 1} = 33 \cdot {0.3939393939393939}^{10 - 1} = 33 \cdot {0.3939393939393939}^{9} = 33 \cdot 0.00022848869217709738 = 0.0075401268418442136$

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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