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 = 0.5405405405405406

r=0.5405405405405406

该系列的和是：

s = 171

s=171

此系列的通用形式是：

a_{n} = 111 \cdot {0.5405405405405406}^{n - 1}

a_n=111*0.5405405405405406^(n-1)

这个序列的第n项是：

111, 60, 32.432432432432435, 17.53104455807159, 9.476240301660319, 5.122292054951523, 2.768806516190013, 1.4966521709135208, 0.8090011734667679, 0.43729793160365843

111,60,32.432432432432435,17.53104455807159,9.476240301660319,5.122292054951523,2.768806516190013,1.4966521709135208,0.8090011734667679,0.43729793160365843

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：

$a_{2} a_{1} = \frac{60}{111} = 0.5405405405405406$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.5405405405405406$

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： $a = 111$ 、公比： $r = 0.5405405405405406$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 111 * ((1 - {0.5405405405405406}^{2}) / (1 - 0.5405405405405406))$

$s_{2} = 111 * ((1 - 0.2921840759678598) / (1 - 0.5405405405405406))$

$s_{2} = 111 * (0.7078159240321402 / (1 - 0.5405405405405406))$

$s_{2} = 111 * (0.7078159240321402 / 0.45945945945945943)$

$s_{2} = 111 \cdot 1.5405405405405406$

$s_{2} = 171$

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： $a = 111$ 和公比： $r = 0.5405405405405406$ 插入几何级数的公式：

$a_{n} = 111 \cdot {0.5405405405405406}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 111$

$a_{2} = a_{1} \cdot r^{n - 1} = 111 \cdot {0.5405405405405406}^{2 - 1} = 111 \cdot {0.5405405405405406}^{1} = 111 \cdot 0.5405405405405406 = 60$

$a_{3} = a_{1} \cdot r^{n - 1} = 111 \cdot {0.5405405405405406}^{3 - 1} = 111 \cdot {0.5405405405405406}^{2} = 111 \cdot 0.2921840759678598 = 32.432432432432435$

$a_{4} = a_{1} \cdot r^{n - 1} = 111 \cdot {0.5405405405405406}^{4 - 1} = 111 \cdot {0.5405405405405406}^{3} = 111 \cdot 0.1579373383610053 = 17.53104455807159$

$a_{5} = a_{1} \cdot r^{n - 1} = 111 \cdot {0.5405405405405406}^{5 - 1} = 111 \cdot {0.5405405405405406}^{4} = 111 \cdot 0.08537153424919205 = 9.476240301660319$

$a_{6} = a_{1} \cdot r^{n - 1} = 111 \cdot {0.5405405405405406}^{6 - 1} = 111 \cdot {0.5405405405405406}^{5} = 111 \cdot 0.046146775269833545 = 5.122292054951523$

$a_{7} = a_{1} \cdot r^{n - 1} = 111 \cdot {0.5405405405405406}^{7 - 1} = 111 \cdot {0.5405405405405406}^{6} = 111 \cdot 0.024944202848558677 = 2.768806516190013$

$a_{8} = a_{1} \cdot r^{n - 1} = 111 \cdot {0.5405405405405406}^{8 - 1} = 111 \cdot {0.5405405405405406}^{7} = 111 \cdot 0.013483352891112799 = 1.4966521709135208$

$a_{9} = a_{1} \cdot r^{n - 1} = 111 \cdot {0.5405405405405406}^{9 - 1} = 111 \cdot {0.5405405405405406}^{8} = 111 \cdot 0.0072882988600609724 = 0.8090011734667679$

$a_{10} = a_{1} \cdot r^{n - 1} = 111 \cdot {0.5405405405405406}^{10 - 1} = 111 \cdot {0.5405405405405406}^{9} = 111 \cdot 0.003939621005438364 = 0.43729793160365843$

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