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解答 - 几何数列

公比是：

r = 0.018518518518518517

r=0.018518518518518517

该系列的和是：

s = 110

s=110

此系列的通用形式是：

a_{n} = 108 \cdot {0.018518518518518517}^{n - 1}

a_n=108*0.018518518518518517^(n-1)

这个序列的第n项是：

108, 2, 0.037037037037037035, 0.0006858710562414265, 1.2701315856322713 E - 05, 2.3520955289486505 E - 07, 4.3557324610160185 E - 09, 8.066171224103738 E - 11, 1.4937354118710625 E - 12, 2.766176688650116 E - 14

108,2,0.037037037037037035,0.0006858710562414265,1.2701315856322713E-05,2.3520955289486505E-07,4.3557324610160185E-09,8.066171224103738E-11,1.4937354118710625E-12,2.766176688650116E-14

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：

$a_{2} a_{1} = \frac{2}{108} = 0.018518518518518517$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.018518518518518517$

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： $a = 108$ 、公比： $r = 0.018518518518518517$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 108 * ((1 - {0.018518518518518517}^{2}) / (1 - 0.018518518518518517))$

$s_{2} = 108 * ((1 - 0.0003429355281207133) / (1 - 0.018518518518518517))$

$s_{2} = 108 * (0.9996570644718793 / (1 - 0.018518518518518517))$

$s_{2} = 108 * (0.9996570644718793 / 0.9814814814814815)$

$s_{2} = 108 \cdot 1.0185185185185186$

$s_{2} = 110.00000000000001$

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： $a = 108$ 和公比： $r = 0.018518518518518517$ 插入几何级数的公式：

$a_{n} = 108 \cdot {0.018518518518518517}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 108$

$a_{2} = a_{1} \cdot r^{n - 1} = 108 \cdot {0.018518518518518517}^{2 - 1} = 108 \cdot {0.018518518518518517}^{1} = 108 \cdot 0.018518518518518517 = 2$

$a_{3} = a_{1} \cdot r^{n - 1} = 108 \cdot {0.018518518518518517}^{3 - 1} = 108 \cdot {0.018518518518518517}^{2} = 108 \cdot 0.0003429355281207133 = 0.037037037037037035$

$a_{4} = a_{1} \cdot r^{n - 1} = 108 \cdot {0.018518518518518517}^{4 - 1} = 108 \cdot {0.018518518518518517}^{3} = 108 \cdot 6.350657928161356 E - 06 = 0.0006858710562414265$

$a_{5} = a_{1} \cdot r^{n - 1} = 108 \cdot {0.018518518518518517}^{5 - 1} = 108 \cdot {0.018518518518518517}^{4} = 108 \cdot 1.1760477644743252 E - 07 = 1.2701315856322713 E - 05$

$a_{6} = a_{1} \cdot r^{n - 1} = 108 \cdot {0.018518518518518517}^{6 - 1} = 108 \cdot {0.018518518518518517}^{5} = 108 \cdot 2.1778662305080097 E - 09 = 2.3520955289486505 E - 07$

$a_{7} = a_{1} \cdot r^{n - 1} = 108 \cdot {0.018518518518518517}^{7 - 1} = 108 \cdot {0.018518518518518517}^{6} = 108 \cdot 4.033085612051869 E - 11 = 4.3557324610160185 E - 09$

$a_{8} = a_{1} \cdot r^{n - 1} = 108 \cdot {0.018518518518518517}^{8 - 1} = 108 \cdot {0.018518518518518517}^{7} = 108 \cdot 7.468677059355313 E - 13 = 8.066171224103738 E - 11$

$a_{9} = a_{1} \cdot r^{n - 1} = 108 \cdot {0.018518518518518517}^{9 - 1} = 108 \cdot {0.018518518518518517}^{8} = 108 \cdot 1.3830883443250578 E - 14 = 1.4937354118710625 E - 12$

$a_{10} = a_{1} \cdot r^{n - 1} = 108 \cdot {0.018518518518518517}^{10 - 1} = 108 \cdot {0.018518518518518517}^{9} = 108 \cdot 2.5612747117130703 E - 16 = 2.766176688650116 E - 14$

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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