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解答 - 几何数列

公比是：

r = 0.967741935483871

r=0.967741935483871

该系列的和是：

s = 60

s=60

此系列的通用形式是：

a_{n} = 31 \cdot {0.967741935483871}^{n - 1}

a_n=31*0.967741935483871^(n-1)

这个序列的第n项是：

31, 30, 29.03225806451613, 28.095733610822062, 27.18941962337619, 26.312341571009217, 25.463556359041178, 24.642151315201144, 23.847243208259172, 23.077977298315325

31,30,29.03225806451613,28.095733610822062,27.18941962337619,26.312341571009217,25.463556359041178,24.642151315201144,23.847243208259172,23.077977298315325

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：

$a_{2} a_{1} = \frac{30}{31} = 0.967741935483871$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.967741935483871$

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： $a = 31$ 、公比： $r = 0.967741935483871$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 31 * ((1 - {0.967741935483871}^{2}) / (1 - 0.967741935483871))$

$s_{2} = 31 * ((1 - 0.9365244536940688) / (1 - 0.967741935483871))$

$s_{2} = 31 * (0.06347554630593122 / (1 - 0.967741935483871))$

$s_{2} = 31 * (0.06347554630593122 / 0.032258064516129004)$

$s_{2} = 31 \cdot 1.9677419354838694$

$s_{2} = 60.99999999999995$

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： $a = 31$ 和公比： $r = 0.967741935483871$ 插入几何级数的公式：

$a_{n} = 31 \cdot {0.967741935483871}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 31$

$a_{2} = a_{1} \cdot r^{n - 1} = 31 \cdot {0.967741935483871}^{2 - 1} = 31 \cdot {0.967741935483871}^{1} = 31 \cdot 0.967741935483871 = 30$

$a_{3} = a_{1} \cdot r^{n - 1} = 31 \cdot {0.967741935483871}^{3 - 1} = 31 \cdot {0.967741935483871}^{2} = 31 \cdot 0.9365244536940688 = 29.03225806451613$

$a_{4} = a_{1} \cdot r^{n - 1} = 31 \cdot {0.967741935483871}^{4 - 1} = 31 \cdot {0.967741935483871}^{3} = 31 \cdot 0.906313987445873 = 28.095733610822062$

$a_{5} = a_{1} \cdot r^{n - 1} = 31 \cdot {0.967741935483871}^{5 - 1} = 31 \cdot {0.967741935483871}^{4} = 31 \cdot 0.8770780523669739 = 27.18941962337619$

$a_{6} = a_{1} \cdot r^{n - 1} = 31 \cdot {0.967741935483871}^{6 - 1} = 31 \cdot {0.967741935483871}^{5} = 31 \cdot 0.8487852119680392 = 26.312341571009217$

$a_{7} = a_{1} \cdot r^{n - 1} = 31 \cdot {0.967741935483871}^{7 - 1} = 31 \cdot {0.967741935483871}^{6} = 31 \cdot 0.821405043840038 = 25.463556359041178$

$a_{8} = a_{1} \cdot r^{n - 1} = 31 \cdot {0.967741935483871}^{8 - 1} = 31 \cdot {0.967741935483871}^{7} = 31 \cdot 0.7949081069419723 = 24.642151315201144$

$a_{9} = a_{1} \cdot r^{n - 1} = 31 \cdot {0.967741935483871}^{9 - 1} = 31 \cdot {0.967741935483871}^{8} = 31 \cdot 0.7692659099438443 = 23.847243208259172$

$a_{10} = a_{1} \cdot r^{n - 1} = 31 \cdot {0.967741935483871}^{10 - 1} = 31 \cdot {0.967741935483871}^{9} = 31 \cdot 0.744450880590817 = 23.077977298315325$

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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