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解答 - 几何数列

公比是：

r = 0.020202020202020204

r=0.020202020202020204

该系列的和是：

s = 100

s=100

此系列的通用形式是：

a_{n} = 99 \cdot {0.020202020202020204}^{n - 1}

a_n=99*0.020202020202020204^(n-1)

这个序列的第n项是：

99, 2, 0.04040404040404041, 0.0008162432404856649, 1.6489762434053837 E - 05, 3.331265138192695 E - 07, 6.729828562005445 E - 09, 1.3595613256576657 E - 10, 2.7465885366821533 E - 12, 5.5486637104689975 E - 14

99,2,0.04040404040404041,0.0008162432404856649,1.6489762434053837E-05,3.331265138192695E-07,6.729828562005445E-09,1.3595613256576657E-10,2.7465885366821533E-12,5.5486637104689975E-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：

$a_{2} a_{1} = \frac{2}{99} = 0.020202020202020204$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.020202020202020204$

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： $a = 99$ 、公比： $r = 0.020202020202020204$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 99 * ((1 - {0.020202020202020204}^{2}) / (1 - 0.020202020202020204))$

$s_{2} = 99 * ((1 - 0.0004081216202428324) / (1 - 0.020202020202020204))$

$s_{2} = 99 * (0.9995918783797572 / (1 - 0.020202020202020204))$

$s_{2} = 99 * (0.9995918783797572 / 0.9797979797979798)$

$s_{2} = 99 \cdot 1.02020202020202$

$s_{2} = 100.99999999999999$

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： $a = 99$ 和公比： $r = 0.020202020202020204$ 插入几何级数的公式：

$a_{n} = 99 \cdot {0.020202020202020204}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 99$

$a_{2} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.020202020202020204}^{2 - 1} = 99 \cdot {0.020202020202020204}^{1} = 99 \cdot 0.020202020202020204 = 2$

$a_{3} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.020202020202020204}^{3 - 1} = 99 \cdot {0.020202020202020204}^{2} = 99 \cdot 0.0004081216202428324 = 0.04040404040404041$

$a_{4} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.020202020202020204}^{4 - 1} = 99 \cdot {0.020202020202020204}^{3} = 99 \cdot 8.244881217026918 E - 06 = 0.0008162432404856649$

$a_{5} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.020202020202020204}^{5 - 1} = 99 \cdot {0.020202020202020204}^{4} = 99 \cdot 1.6656325690963473 E - 07 = 1.6489762434053837 E - 05$

$a_{6} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.020202020202020204}^{6 - 1} = 99 \cdot {0.020202020202020204}^{5} = 99 \cdot 3.364914281002722 E - 09 = 3.331265138192695 E - 07$

$a_{7} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.020202020202020204}^{7 - 1} = 99 \cdot {0.020202020202020204}^{6} = 99 \cdot 6.797806628288329 E - 11 = 6.729828562005445 E - 09$

$a_{8} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.020202020202020204}^{8 - 1} = 99 \cdot {0.020202020202020204}^{7} = 99 \cdot 1.3732942683410765 E - 12 = 1.3595613256576657 E - 10$

$a_{9} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.020202020202020204}^{9 - 1} = 99 \cdot {0.020202020202020204}^{8} = 99 \cdot 2.774331855234498 E - 14 = 2.7465885366821533 E - 12$

$a_{10} = a_{1} \cdot r^{n - 1} = 99 \cdot {0.020202020202020204}^{10 - 1} = 99 \cdot {0.020202020202020204}^{9} = 99 \cdot 5.604710818655553 E - 16 = 5.5486637104689975 E - 14$

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为什么学习这个

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