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解答 - 几何数列

公比是：

r = 0.13636363636363635

r=0.13636363636363635

该系列的和是：

s = 25

s=25

此系列的通用形式是：

a_{n} = 22 \cdot {0.13636363636363635}^{n - 1}

a_n=22*0.13636363636363635^(n-1)

这个序列的第n项是：

22, 3, 0.409090909090909, 0.05578512396694213, 0.007607062359128473, 0.0010373266853357006, 0.00014145363890941373, 1.9289132578556414 E - 05, 2.630336260712238 E - 06, 3.5868221736985066 E - 07

22,3,0.409090909090909,0.05578512396694213,0.007607062359128473,0.0010373266853357006,0.00014145363890941373,1.9289132578556414E-05,2.630336260712238E-06,3.5868221736985066E-07

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：

$a_{2} a_{1} = \frac{3}{22} = 0.13636363636363635$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.13636363636363635$

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： $r = 0.13636363636363635$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 22 * ((1 - {0.13636363636363635}^{2}) / (1 - 0.13636363636363635))$

$s_{2} = 22 * ((1 - 0.018595041322314047) / (1 - 0.13636363636363635))$

$s_{2} = 22 * (0.981404958677686 / (1 - 0.13636363636363635))$

$s_{2} = 22 * (0.981404958677686 / 0.8636363636363636)$

$s_{2} = 22 \cdot 1.1363636363636365$

$s_{2} = 25.000000000000004$

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： $r = 0.13636363636363635$ 插入几何级数的公式：

$a_{n} = 22 \cdot {0.13636363636363635}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 22$

$a_{2} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.13636363636363635}^{2 - 1} = 22 \cdot {0.13636363636363635}^{1} = 22 \cdot 0.13636363636363635 = 3$

$a_{3} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.13636363636363635}^{3 - 1} = 22 \cdot {0.13636363636363635}^{2} = 22 \cdot 0.018595041322314047 = 0.409090909090909$

$a_{4} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.13636363636363635}^{4 - 1} = 22 \cdot {0.13636363636363635}^{3} = 22 \cdot 0.002535687453042824 = 0.05578512396694213$

$a_{5} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.13636363636363635}^{5 - 1} = 22 \cdot {0.13636363636363635}^{4} = 22 \cdot 0.00034577556177856696 = 0.007607062359128473$

$a_{6} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.13636363636363635}^{6 - 1} = 22 \cdot {0.13636363636363635}^{5} = 22 \cdot 4.7151212969804575 E - 05 = 0.0010373266853357006$

$a_{7} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.13636363636363635}^{7 - 1} = 22 \cdot {0.13636363636363635}^{6} = 22 \cdot 6.429710859518805 E - 06 = 0.00014145363890941373$

$a_{8} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.13636363636363635}^{8 - 1} = 22 \cdot {0.13636363636363635}^{7} = 22 \cdot 8.767787535707462 E - 07 = 1.9289132578556414 E - 05$

$a_{9} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.13636363636363635}^{9 - 1} = 22 \cdot {0.13636363636363635}^{8} = 22 \cdot 1.1956073912328355 E - 07 = 2.630336260712238 E - 06$

$a_{10} = a_{1} \cdot r^{n - 1} = 22 \cdot {0.13636363636363635}^{10 - 1} = 22 \cdot {0.13636363636363635}^{9} = 22 \cdot 1.630373715317503 E - 08 = 3.5868221736985066 E - 07$

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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