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解答 - 几何数列

公比是：

r = 0.9704704704704705

r=0.9704704704704705

该系列的和是：

s = 3936

s=3936

此系列的通用形式是：

a_{n} = 1998 \cdot {0.9704704704704705}^{n - 1}

a_n=1998*0.9704704704704705^(n-1)

这个序列的第n项是：

1998, 1939, 1881.7422422422424, 1826.1752791329873, 1772.2491823017328, 1719.9154977392693, 1669.1272022604821, 1619.8386612527904, 1572.0055876722527, 1525.5850022504997

1998,1939,1881.7422422422424,1826.1752791329873,1772.2491823017328,1719.9154977392693,1669.1272022604821,1619.8386612527904,1572.0055876722527,1525.5850022504997

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：

$a_{2} a_{1} = \frac{1939}{1998} = 0.9704704704704705$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.9704704704704705$

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： $a = 1, 998$ 、公比： $r = 0.9704704704704705$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 1998 * ((1 - {0.9704704704704705}^{2}) / (1 - 0.9704704704704705))$

$s_{2} = 1998 * ((1 - 0.9418129340551764) / (1 - 0.9704704704704705))$

$s_{2} = 1998 * (0.05818706594482359 / (1 - 0.9704704704704705))$

$s_{2} = 1998 * (0.05818706594482359 / 0.029529529529529475)$

$s_{2} = 1998 \cdot 1.9704704704704703$

$s_{2} = 3936.9999999999995$

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： $a = 1, 998$ 和公比： $r = 0.9704704704704705$ 插入几何级数的公式：

$a_{n} = 1998 \cdot {0.9704704704704705}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 1998$

$a_{2} = a_{1} \cdot r^{n - 1} = 1998 \cdot {0.9704704704704705}^{2 - 1} = 1998 \cdot {0.9704704704704705}^{1} = 1998 \cdot 0.9704704704704705 = 1939$

$a_{3} = a_{1} \cdot r^{n - 1} = 1998 \cdot {0.9704704704704705}^{3 - 1} = 1998 \cdot {0.9704704704704705}^{2} = 1998 \cdot 0.9418129340551764 = 1881.7422422422424$

$a_{4} = a_{1} \cdot r^{n - 1} = 1998 \cdot {0.9704704704704705}^{4 - 1} = 1998 \cdot {0.9704704704704705}^{3} = 1998 \cdot 0.9140016412077013 = 1826.1752791329873$

$a_{5} = a_{1} \cdot r^{n - 1} = 1998 \cdot {0.9704704704704705}^{5 - 1} = 1998 \cdot {0.9704704704704705}^{4} = 1998 \cdot 0.88701160275362 = 1772.2491823017328$

$a_{6} = a_{1} \cdot r^{n - 1} = 1998 \cdot {0.9704704704704705}^{6 - 1} = 1998 \cdot {0.9704704704704705}^{5} = 1998 \cdot 0.8608185674370717 = 1719.9154977392693$

$a_{7} = a_{1} \cdot r^{n - 1} = 1998 \cdot {0.9704704704704705}^{7 - 1} = 1998 \cdot {0.9704704704704705}^{6} = 1998 \cdot 0.8353990001303715 = 1669.1272022604821$

$a_{8} = a_{1} \cdot r^{n - 1} = 1998 \cdot {0.9704704704704705}^{8 - 1} = 1998 \cdot {0.9704704704704705}^{7} = 1998 \cdot 0.8107300606870823 = 1619.8386612527904$

$a_{9} = a_{1} \cdot r^{n - 1} = 1998 \cdot {0.9704704704704705}^{9 - 1} = 1998 \cdot {0.9704704704704705}^{8} = 1998 \cdot 0.7867895834195459 = 1572.0055876722527$

$a_{10} = a_{1} \cdot r^{n - 1} = 1998 \cdot {0.9704704704704705}^{10 - 1} = 1998 \cdot {0.9704704704704705}^{9} = 1998 \cdot 0.7635560571824322 = 1525.5850022504997$

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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