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解答 - 几何数列

公比是：

r = 0.09090909090909091

r=0.09090909090909091

该系列的和是：

s = 24

s=24

此系列的通用形式是：

a_{n} = 22 \cdot {0.09090909090909091}^{n - 1}

a_n=22*0.09090909090909091^(n-1)

这个序列的第n项是：

22, 2, 0.18181818181818182, 0.01652892561983471, 0.0015026296018031556, 0.00013660269107301415, 1.2418426461183106 E - 05, 1.128947860107555 E - 06, 1.0263162364614137 E - 07, 9.330147604194671 E - 09

22,2,0.18181818181818182,0.01652892561983471,0.0015026296018031556,0.00013660269107301415,1.2418426461183106E-05,1.128947860107555E-06,1.0263162364614137E-07,9.330147604194671E-09

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：

$a_{2} a_{1} = \frac{2}{22} = 0.09090909090909091$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.09090909090909091$

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： $a = 22$ 、公比： $r = 0.09090909090909091$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 22 * ((1 - {0.09090909090909091}^{2}) / (1 - 0.09090909090909091))$

$s_{2} = 22 * ((1 - 0.008264462809917356) / (1 - 0.09090909090909091))$

$s_{2} = 22 * (0.9917355371900827 / (1 - 0.09090909090909091))$

$s_{2} = 22 * (0.9917355371900827 / 0.9090909090909091)$

$s_{2} = 22 \cdot 1.090909090909091$

$s_{2} = 24.000000000000004$

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： $a = 22$ 和公比： $r = 0.09090909090909091$ 插入几何级数的公式：

$a_{n} = 22 \cdot {0.09090909090909091}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 22$

$a_{2} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.09090909090909091}^{2 - 1} = 22 \cdot {0.09090909090909091}^{1} = 22 \cdot 0.09090909090909091 = 2$

$a_{3} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.09090909090909091}^{3 - 1} = 22 \cdot {0.09090909090909091}^{2} = 22 \cdot 0.008264462809917356 = 0.18181818181818182$

$a_{4} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.09090909090909091}^{4 - 1} = 22 \cdot {0.09090909090909091}^{3} = 22 \cdot 0.0007513148009015778 = 0.01652892561983471$

$a_{5} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.09090909090909091}^{5 - 1} = 22 \cdot {0.09090909090909091}^{4} = 22 \cdot 6.830134553650708 E - 05 = 0.0015026296018031556$

$a_{6} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.09090909090909091}^{6 - 1} = 22 \cdot {0.09090909090909091}^{5} = 22 \cdot 6.209213230591552 E - 06 = 0.00013660269107301415$

$a_{7} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.09090909090909091}^{7 - 1} = 22 \cdot {0.09090909090909091}^{6} = 22 \cdot 5.644739300537775 E - 07 = 1.2418426461183106 E - 05$

$a_{8} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.09090909090909091}^{8 - 1} = 22 \cdot {0.09090909090909091}^{7} = 22 \cdot 5.1315811823070687 E - 08 = 1.128947860107555 E - 06$

$a_{9} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.09090909090909091}^{9 - 1} = 22 \cdot {0.09090909090909091}^{8} = 22 \cdot 4.665073802097335 E - 09 = 1.0263162364614137 E - 07$

$a_{10} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.09090909090909091}^{10 - 1} = 22 \cdot {0.09090909090909091}^{9} = 22 \cdot 4.2409761837248504 E - 10 = 9.330147604194671 E - 09$

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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