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解答 - 几何数列

公比是：

r = 0.999989999899999

r=0.999989999899999

该系列的和是：

s = 199996

s=199996

此系列的通用形式是：

a_{n} = 99999 \cdot {0.999989999899999}^{n - 1}

a_n=99999*0.999989999899999^(n-1)

这个序列的第n项是：

99999, 99998, 99997.0000100001, 99996.00003000021, 99995.00006000022, 99994.00010000002, 99993.00014999953, 99992.00020999862, 99991.00027999723, 99990.00035999523

99999,99998,99997.0000100001,99996.00003000021,99995.00006000022,99994.00010000002,99993.00014999953,99992.00020999862,99991.00027999723,99990.00035999523

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：

$a_{2} a_{1} = \frac{99998}{99999} = 0.999989999899999$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.999989999899999$

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： $a = 99, 999$ 、公比： $r = 0.999989999899999$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 99999 * ((1 - {0.999989999899999}^{2}) / (1 - 0.999989999899999))$

$s_{2} = 99999 * ((1 - 0.9999799999000001) / (1 - 0.999989999899999))$

$s_{2} = 99999 * (2.0000099999917254 E - 05 / (1 - 0.999989999899999))$

$s_{2} = 99999 * (2.0000099999917254 E - 05 / 1.0000100000961964 E - 05)$

$s_{2} = 99999 \cdot 1.9999899998993345$

$s_{2} = 199996.99999993356$

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： $a = 99, 999$ 和公比： $r = 0.999989999899999$ 插入几何级数的公式：

$a_{n} = 99999 \cdot {0.999989999899999}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 99999$

$a_{2} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.999989999899999}^{2 - 1} = 99999 \cdot {0.999989999899999}^{1} = 99999 \cdot 0.999989999899999 = 99998$

$a_{3} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.999989999899999}^{3 - 1} = 99999 \cdot {0.999989999899999}^{2} = 99999 \cdot 0.9999799999000001 = 99997.0000100001$

$a_{4} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.999989999899999}^{4 - 1} = 99999 \cdot {0.999989999899999}^{3} = 99999 \cdot 0.9999700000000021 = 99996.00003000021$

$a_{5} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.999989999899999}^{5 - 1} = 99999 \cdot {0.999989999899999}^{4} = 99999 \cdot 0.9999600002000042 = 99995.00006000022$

$a_{6} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.999989999899999}^{6 - 1} = 99999 \cdot {0.999989999899999}^{5} = 99999 \cdot 0.9999500005000052 = 99994.00010000002$

$a_{7} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.999989999899999}^{7 - 1} = 99999 \cdot {0.999989999899999}^{6} = 99999 \cdot 0.9999400009000042 = 99993.00014999953$

$a_{8} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.999989999899999}^{8 - 1} = 99999 \cdot {0.999989999899999}^{7} = 99999 \cdot 0.9999300014000002 = 99992.00020999862$

$a_{9} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.999989999899999}^{9 - 1} = 99999 \cdot {0.999989999899999}^{8} = 99999 \cdot 0.9999200019999923 = 99991.00027999723$

$a_{10} = a_{1} \cdot r^{n - 1} = 99999 \cdot {0.999989999899999}^{10 - 1} = 99999 \cdot {0.999989999899999}^{9} = 99999 \cdot 0.9999100026999793 = 99990.00035999523$

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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