输入一个方程或问题

无法识别摄像头输入！

解答 - 几何数列

公比是：

r = 0.7272727272727273

r=0.7272727272727273

该系列的和是：

s = 190

s=190

此系列的通用形式是：

a_{n} = 110 \cdot {0.7272727272727273}^{n - 1}

a_n=110*0.7272727272727273^(n-1)

这个序列的第n项是：

110, 80, 58.18181818181819, 42.31404958677686, 30.77385424492863, 22.380984905402638, 16.27707993120192, 11.837876313601397, 8.609364591710106, 6.2613560666982595

110,80,58.18181818181819,42.31404958677686,30.77385424492863,22.380984905402638,16.27707993120192,11.837876313601397,8.609364591710106,6.2613560666982595

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：

$a_{2} a_{1} = \frac{80}{110} = 0.7272727272727273$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.7272727272727273$

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： $a = 110$ 、公比： $r = 0.7272727272727273$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 110 * ((1 - {0.7272727272727273}^{2}) / (1 - 0.7272727272727273))$

$s_{2} = 110 * ((1 - 0.5289256198347108) / (1 - 0.7272727272727273))$

$s_{2} = 110 * (0.47107438016528924 / (1 - 0.7272727272727273))$

$s_{2} = 110 * (0.47107438016528924 / 0.2727272727272727)$

$s_{2} = 110 \cdot 1.7272727272727273$

$s_{2} = 190$

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： $a = 110$ 和公比： $r = 0.7272727272727273$ 插入几何级数的公式：

$a_{n} = 110 \cdot {0.7272727272727273}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 110$

$a_{2} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.7272727272727273}^{2 - 1} = 110 \cdot {0.7272727272727273}^{1} = 110 \cdot 0.7272727272727273 = 80$

$a_{3} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.7272727272727273}^{3 - 1} = 110 \cdot {0.7272727272727273}^{2} = 110 \cdot 0.5289256198347108 = 58.18181818181819$

$a_{4} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.7272727272727273}^{4 - 1} = 110 \cdot {0.7272727272727273}^{3} = 110 \cdot 0.38467317806160783 = 42.31404958677686$

$a_{5} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.7272727272727273}^{5 - 1} = 110 \cdot {0.7272727272727273}^{4} = 110 \cdot 0.279762311317533 = 30.77385424492863$

$a_{6} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.7272727272727273}^{6 - 1} = 110 \cdot {0.7272727272727273}^{5} = 110 \cdot 0.20346349914002398 = 22.380984905402638$

$a_{7} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.7272727272727273}^{7 - 1} = 110 \cdot {0.7272727272727273}^{6} = 110 \cdot 0.14797345392001746 = 16.27707993120192$

$a_{8} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.7272727272727273}^{8 - 1} = 110 \cdot {0.7272727272727273}^{7} = 110 \cdot 0.10761705739637634 = 11.837876313601397$

$a_{9} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.7272727272727273}^{9 - 1} = 110 \cdot {0.7272727272727273}^{8} = 110 \cdot 0.07826695083372824 = 8.609364591710106$

$a_{10} = a_{1} \cdot r^{n - 1} = 110 \cdot {0.7272727272727273}^{10 - 1} = 110 \cdot {0.7272727272727273}^{9} = 110 \cdot 0.056921418788166 = 6.2613560666982595$

我们做得怎么样？

给我们反馈

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。

术语和主题

几何数列