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解答 - 几何数列

公比是：

r = 0.08108108108108109

r=0.08108108108108109

该系列的和是：

s = 40

s=40

此系列的通用形式是：

a_{n} = 37 \cdot {0.08108108108108109}^{n - 1}

a_n=37*0.08108108108108109^(n-1)

这个序列的第n项是：

37, 3, 0.24324324324324326, 0.019722425127830536, 0.0015991155509051786, 0.00012965801764096044, 1.0512812241158955 E - 05, 8.52390181715591 E - 07, 6.911271743639929 E - 08, 5.603733846194536 E - 09

37,3,0.24324324324324326,0.019722425127830536,0.0015991155509051786,0.00012965801764096044,1.0512812241158955E-05,8.52390181715591E-07,6.911271743639929E-08,5.603733846194536E-09

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：

$a_{2} a_{1} = \frac{3}{37} = 0.08108108108108109$

该序列的公比（ $r$ ）保持不变，并且等于两个连续项的商。
$r = 0.08108108108108109$

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： $a = 37$ 、公比： $r = 0.08108108108108109$ 和元素数目 $n = 2$ 插入几何级数求和公式：

$s_{2} = 37 * ((1 - {0.08108108108108109}^{2}) / (1 - 0.08108108108108109))$

$s_{2} = 37 * ((1 - 0.006574141709276845) / (1 - 0.08108108108108109))$

$s_{2} = 37 * (0.9934258582907232 / (1 - 0.08108108108108109))$

$s_{2} = 37 * (0.9934258582907232 / 0.9189189189189189)$

$s_{2} = 37 \cdot 1.0810810810810811$

$s_{2} = 40$

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： $a = 37$ 和公比： $r = 0.08108108108108109$ 插入几何级数的公式：

$a_{n} = 37 \cdot {0.08108108108108109}^{n - 1}$

4. 找到第n项

使用通用公式找到第n项

$a_{1} = 37$

$a_{2} = a_{1} \cdot r^{n - 1} = 37 \cdot {0.08108108108108109}^{2 - 1} = 37 \cdot {0.08108108108108109}^{1} = 37 \cdot 0.08108108108108109 = 3$

$a_{3} = a_{1} \cdot r^{n - 1} = 37 \cdot {0.08108108108108109}^{3 - 1} = 37 \cdot {0.08108108108108109}^{2} = 37 \cdot 0.006574141709276845 = 0.24324324324324326$

$a_{4} = a_{1} \cdot r^{n - 1} = 37 \cdot {0.08108108108108109}^{4 - 1} = 37 \cdot {0.08108108108108109}^{3} = 37 \cdot 0.0005330385169683929 = 0.019722425127830536$

$a_{5} = a_{1} \cdot r^{n - 1} = 37 \cdot {0.08108108108108109}^{5 - 1} = 37 \cdot {0.08108108108108109}^{4} = 37 \cdot 4.321933921365348 E - 05 = 0.0015991155509051786$

$a_{6} = a_{1} \cdot r^{n - 1} = 37 \cdot {0.08108108108108109}^{6 - 1} = 37 \cdot {0.08108108108108109}^{5} = 37 \cdot 3.504270747052985 E - 06 = 0.00012965801764096044$

$a_{7} = a_{1} \cdot r^{n - 1} = 37 \cdot {0.08108108108108109}^{7 - 1} = 37 \cdot {0.08108108108108109}^{6} = 37 \cdot 2.8413006057186365 E - 07 = 1.0512812241158955 E - 05$

$a_{8} = a_{1} \cdot r^{n - 1} = 37 \cdot {0.08108108108108109}^{8 - 1} = 37 \cdot {0.08108108108108109}^{7} = 37 \cdot 2.3037572478799757 E - 08 = 8.52390181715591 E - 07$

$a_{9} = a_{1} \cdot r^{n - 1} = 37 \cdot {0.08108108108108109}^{9 - 1} = 37 \cdot {0.08108108108108109}^{8} = 37 \cdot 1.8679112820648454 E - 09 = 6.911271743639929 E - 08$

$a_{10} = a_{1} \cdot r^{n - 1} = 37 \cdot {0.08108108108108109}^{10 - 1} = 37 \cdot {0.08108108108108109}^{9} = 37 \cdot 1.5145226611336583 E - 10 = 5.603733846194536 E - 09$

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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