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解答 - 几何数列

公比是：

r = 0.00999009990099901

r=0.00999009990099901

该系列的和是：

s = 100998

s=100998

此系列的通用形式是：

a_{n} = 99999 \cdot {0.00999009990099901}^{n - 1}

a_n=99999*0.00999009990099901^(n-1)

这个序列的第n项是：

99999, 999, 9.980109801098012, 0.0997022939359085, 0.0009960358767784937, 9.950497913996293 E - 06, 9.940646822550522 E - 08, 9.93080548378281 E - 10, 9.920973888037907 E - 12, 9.911152025670127 E - 14

99999,999,9.980109801098012,0.0997022939359085,0.0009960358767784937,9.950497913996293E-06,9.940646822550522E-08,9.93080548378281E-10,9.920973888037907E-12,9.911152025670127E-14

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：

$a_{2} a_{1} = \frac{999}{99999} = 0.00999009990099901$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.00999009990099901$

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： $a = 99, 999$ 、公比： $r = 0.00999009990099901$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 99999 * ((1 - {0.00999009990099901}^{2}) / (1 - 0.00999009990099901))$

$s_{2} = 99999 * ((1 - 9.980209603194043 E - 05) / (1 - 0.00999009990099901))$

$s_{2} = 99999 * (0.9999001979039681 / (1 - 0.00999009990099901))$

$s_{2} = 99999 * (0.9999001979039681 / 0.990009900099001)$

$s_{2} = 99999 \cdot 1.009990099900999$

$s_{2} = 100998$

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： $a = 99, 999$ 和公比： $r = 0.00999009990099901$ 插入几何级数的公式：

$a_{n} = 99999 \cdot {0.00999009990099901}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 99999$

$a_{2} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.00999009990099901}^{2 - 1} = 99999 \cdot {0.00999009990099901}^{1} = 99999 \cdot 0.00999009990099901 = 999$

$a_{3} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.00999009990099901}^{3 - 1} = 99999 \cdot {0.00999009990099901}^{2} = 99999 \cdot 9.980209603194043 E - 05 = 9.980109801098012$

$a_{4} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.00999009990099901}^{4 - 1} = 99999 \cdot {0.00999009990099901}^{3} = 99999 \cdot 9.970329096881819 E - 07 = 0.0997022939359085$

$a_{5} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.00999009990099901}^{5 - 1} = 99999 \cdot {0.00999009990099901}^{4} = 99999 \cdot 9.96045837236866 E - 09 = 0.0009960358767784937$

$a_{6} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.00999009990099901}^{6 - 1} = 99999 \cdot {0.00999009990099901}^{5} = 99999 \cdot 9.950597419970493 E - 11 = 9.950497913996293 E - 06$

$a_{7} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.00999009990099901}^{7 - 1} = 99999 \cdot {0.00999009990099901}^{6} = 99999 \cdot 9.940746230012823 E - 13 = 9.940646822550522 E - 08$

$a_{8} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.00999009990099901}^{8 - 1} = 99999 \cdot {0.00999009990099901}^{7} = 99999 \cdot 9.930904792830739 E - 15 = 9.93080548378281 E - 10$

$a_{9} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.00999009990099901}^{9 - 1} = 99999 \cdot {0.00999009990099901}^{8} = 99999 \cdot 9.921073098768895 E - 17 = 9.920973888037907 E - 12$

$a_{10} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.00999009990099901}^{10 - 1} = 99999 \cdot {0.00999009990099901}^{9} = 99999 \cdot 9.911251138181508 E - 19 = 9.911152025670127 E - 14$

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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