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解答 - 几何数列

公比是：

r = 0.8888888888888888

r=0.8888888888888888

该系列的和是：

s = 288

s=288

此系列的通用形式是：

a_{n} = 153 \cdot {0.8888888888888888}^{n - 1}

a_n=153*0.8888888888888888^(n-1)

这个序列的第n项是：

153, 136, 120.88888888888889, 107.45679012345677, 95.51714677640602, 84.90413046791646, 75.47033819370351, 67.08474506106978, 59.63088449872869, 53.00523066553661

153,136,120.88888888888889,107.45679012345677,95.51714677640602,84.90413046791646,75.47033819370351,67.08474506106978,59.63088449872869,53.00523066553661

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：

$a_{2} a_{1} = \frac{136}{153} = 0.8888888888888888$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.8888888888888888$

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： $a = 153$ 、公比： $r = 0.8888888888888888$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 153 * ((1 - {0.8888888888888888}^{2}) / (1 - 0.8888888888888888))$

$s_{2} = 153 * ((1 - 0.7901234567901234) / (1 - 0.8888888888888888))$

$s_{2} = 153 * (0.2098765432098766 / (1 - 0.8888888888888888))$

$s_{2} = 153 * (0.2098765432098766 / 0.11111111111111116)$

$s_{2} = 153 \cdot 1.8888888888888884$

$s_{2} = 288.99999999999994$

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： $a = 153$ 和公比： $r = 0.8888888888888888$ 插入几何级数的公式：

$a_{n} = 153 \cdot {0.8888888888888888}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 153$

$a_{2} = a_{1} \cdot r^{n - 1} = 153 \cdot {0.8888888888888888}^{2 - 1} = 153 \cdot {0.8888888888888888}^{1} = 153 \cdot 0.8888888888888888 = 136$

$a_{3} = a_{1} \cdot r^{n - 1} = 153 \cdot {0.8888888888888888}^{3 - 1} = 153 \cdot {0.8888888888888888}^{2} = 153 \cdot 0.7901234567901234 = 120.88888888888889$

$a_{4} = a_{1} \cdot r^{n - 1} = 153 \cdot {0.8888888888888888}^{4 - 1} = 153 \cdot {0.8888888888888888}^{3} = 153 \cdot 0.7023319615912207 = 107.45679012345677$

$a_{5} = a_{1} \cdot r^{n - 1} = 153 \cdot {0.8888888888888888}^{5 - 1} = 153 \cdot {0.8888888888888888}^{4} = 153 \cdot 0.624295076969974 = 95.51714677640602$

$a_{6} = a_{1} \cdot r^{n - 1} = 153 \cdot {0.8888888888888888}^{6 - 1} = 153 \cdot {0.8888888888888888}^{5} = 153 \cdot 0.5549289573066435 = 84.90413046791646$

$a_{7} = a_{1} \cdot r^{n - 1} = 153 \cdot {0.8888888888888888}^{7 - 1} = 153 \cdot {0.8888888888888888}^{6} = 153 \cdot 0.49327018427257197 = 75.47033819370351$

$a_{8} = a_{1} \cdot r^{n - 1} = 153 \cdot {0.8888888888888888}^{8 - 1} = 153 \cdot {0.8888888888888888}^{7} = 153 \cdot 0.43846238602006393 = 67.08474506106978$

$a_{9} = a_{1} \cdot r^{n - 1} = 153 \cdot {0.8888888888888888}^{9 - 1} = 153 \cdot {0.8888888888888888}^{8} = 153 \cdot 0.3897443431289457 = 59.63088449872869$

$a_{10} = a_{1} \cdot r^{n - 1} = 153 \cdot {0.8888888888888888}^{10 - 1} = 153 \cdot {0.8888888888888888}^{9} = 153 \cdot 0.34643941611461837 = 53.00523066553661$

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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